05 Extended Example 2.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "aa37e5c4-422d-46b1-862e-66288f3369c4",
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import *   \n",
    "from libaldo_polyclass  import *\n",
    "from libaldo_lib import *\n",
    "from libaldo__util import *\n",
    "import lib_algebraEq as ae\n",
    "init_printing()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b67dd10b-53e0-4464-ac0f-263c7591aa7a",
   "metadata": {
    "tags": []
   },
   "source": [
    " <img src=\"images/linep.png\" alt=\"ml\" style=\"width : 100%;\"/>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9ec5d316-43be-4a1a-887b-c2e4a1493bfd",
   "metadata": {
    "tags": []
   },
   "source": [
    " <img src=\"images/p21.png\" alt=\"ml\" style=\"width : 100%;\"/>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c8483899-2040-49a4-b7f4-14a801def20c",
   "metadata": {},
   "source": [
    " <img src=\"images/lines.png\" alt=\"ml\" style=\"width : 100%;\"/>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "28ce3196-174b-4a35-801b-a103fa2f1536",
   "metadata": {},
   "outputs": [],
   "source": [
    "x,y,a,b,c,k =symbols('x y a b c k')\n",
    "def f(w):\n",
    "    return(kpow(w-1,2)/w+2/w)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9d4f3648-6c9d-497b-ba18-da25a75b58ba",
   "metadata": {
    "tags": []
   },
   "source": [
    " <img src=\"images/p21a.png\" alt=\"ml\" style=\"width : 100%;\"/>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "da8c1a90-0fef-4a0f-965f-baa7c44cf110",
   "metadata": {},
   "outputs": [
    {
     "data": {
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      "text/latex": [
       "$\\displaystyle \\frac{\\left(c - 1\\right)^{2}}{c} + \\frac{2}{c} + \\frac{\\left(b - 1\\right)^{2}}{b} + \\frac{2}{b} + \\frac{\\left(a - 1\\right)^{2}}{a} + \\frac{2}{a}$"
      ],
      "text/plain": [
       "       2              2              2    \n",
       "(c - 1)    2   (b - 1)    2   (a - 1)    2\n",
       "──────── + ─ + ──────── + ─ + ──────── + ─\n",
       "   c       c      b       b      a       a"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e1=f(a)+f(b)+f(c) ;e1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "15059a4a-7612-4be0-979e-a5d2ce60d8d3",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle if\\;\\frac{\\left(c - 1\\right)^{2}}{c} + \\frac{2}{c} + \\frac{\\left(b - 1\\right)^{2}}{b} + \\frac{2}{b} + \\frac{\\left(a - 1\\right)^{2}}{a} + \\frac{2}{a}\\;=\\;\\frac{2}{c} + \\frac{2}{b} + \\frac{2}{a}\\;$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle then\\;\\frac{\\left(c - 1\\right)^{2}}{c} + \\frac{\\left(b - 1\\right)^{2}}{b} + \\frac{\\left(a - 1\\right)^{2}}{a}\\;=\\;0\\;$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sE(['if ',e1,'=',2*(1/a+1/b+1/c)])\n",
    "sE(['then',e1-2/a-2/b-2/c,'=',0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "f9ee3aef-6947-4e41-8566-9d7c566752a9",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle And..\\;\\frac{\\left(a - 1\\right)^{2}}{a}\\;=\\;0\\;,\\;\\frac{\\left(b - 1\\right)^{2}}{b}\\;=\\;0\\;,\\;\\frac{\\left(c - 1\\right)^{2}}{c}\\;=\\;0\\;$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "e2=e1-2/a-2/b-2/c \n",
    "sE(['And..',part(e2,[0]),'=',0,',',part(e2,[1]),'=',0,',',part(e2,[2]),'=',0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "17b5836a-076f-455a-85df-accbbbf3334e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle Finaly\\;..\\;\\frac{\\left(a - 1\\right)^{2}}{a}\\;=\\;0\\;,\\;\\frac{\\left(b - 1\\right)^{2}}{b}\\;=\\;0\\;,\\;\\frac{\\left(c - 1\\right)^{2}}{c}\\;=\\;0\\;$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle and\\;solve\\;that\\;..a\\;=\\;1\\;,b=\\;1\\;,c\\;=\\;1\\;$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sE(['Finaly ..',part(e2,[0]),'=',0,',',part(e2,[1]),'=',0,',',part(e2,[2]),'=',0])\n",
    "sE(['and solve that ..a =',1,',b=',1,',c =',1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "ab6a5039-47ae-4e52-8e7e-49ef664e2910",
   "metadata": {},
   "outputs": [],
   "source": [
    "def f3(n):\n",
    "    kk=kpow(a,n)+kpow(b,n)+kpow(c,n)\n",
    "    return(kpow(kk,n))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "bff13755-34f1-471a-85b2-33416750e348",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{\\left(a^{3} + b^{3} + c^{3}\\right)^{3} \\left(a^{6} + b^{6} + c^{6}\\right)^{6}}{\\left(a^{9} + b^{9} + c^{9}\\right)^{9}}$"
      ],
      "text/plain": [
       "              3               6\n",
       "⎛ 3    3    3⎞  ⎛ 6    6    6⎞ \n",
       "⎝a  + b  + c ⎠ ⋅⎝a  + b  + c ⎠ \n",
       "───────────────────────────────\n",
       "                      9        \n",
       "        ⎛ 9    9    9⎞         \n",
       "        ⎝a  + b  + c ⎠         "
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M=f3(3)*f3(6)/f3(9);M"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "9b7525b5-96ea-41c6-9629-58927e7d22a8",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{\\left(a^{3} + b^{3} + c^{3}\\right)^{3} \\left(a^{6} + b^{6} + c^{6}\\right)^{6}}{\\left(a^{9} + b^{9} + c^{9}\\right)^{9}}\\;=\\;1\\;$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sE([M,'=',M.subs({a:1,b:1,c:1})])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "d8e74439-2516-4872-a8e8-ef2ca2c0fd06",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle M=\\;1\\;$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sE(['M=',1])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b8f6d541-e6a4-475e-b2b8-d53c369c49d9",
   "metadata": {},
   "source": [
    " <img src=\"images/linep.png\" alt=\"ml\" style=\"width : 100%;\"/>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8e9f1977-4ed5-47f8-83c4-5095469c7b4c",
   "metadata": {
    "tags": []
   },
   "source": [
    " <img src=\"images/p28.png\" alt=\"ml\" style=\"width : 100%;\"/>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "35849418-d861-4c04-9431-df3e5fdb3faa",
   "metadata": {},
   "source": [
    " <img src=\"images/lines.png\" alt=\"ml\" style=\"width : 100%;\"/>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "57d61efe-c0fe-46a8-a7e2-fb22ac29b721",
   "metadata": {},
   "outputs": [],
   "source": [
    "e1,e2,e3= (a*a-b*c)/(b*c),(b*b-a*c)/(a*c),(c*c-a*b)/(a*b) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "64f9eb3a-52b2-40f9-a9ba-c7255a3fe807",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{a^{2} - b c}{b c} + \\frac{- a c + b^{2}}{a c} + \\frac{- a b + c^{2}}{a b}=0$"
      ],
      "text/plain": [
       "(a**2 - b*c)/(b*c) + (-a*c + b**2)/(a*c) + (-a*b + c**2)/(a*b)=0"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P=polyclass(e1+e2+e3,0);P.Q"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "2bc7c0fd-76d8-483c-b217-4baf2002f0aa",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{\\left(a + b + c\\right) \\left(a^{2} - a b - a c + b^{2} - b c + c^{2}\\right)}{a b c}=0$"
      ],
      "text/plain": [
       "(a + b + c)*(a**2 - a*b - a*c + b**2 - b*c + c**2)/(a*b*c)=0"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P.opemat('f',1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "f62e9ac5-c5ed-4cc1-8cb1-eeab9b5b682c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle a + b + c$"
      ],
      "text/plain": [
       "a + b + c"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "part(P.Q,[0,3])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "b34fea7e-674d-44bb-9563-e60a74d8cd83",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle a^{2} - a b - a c + b^{2} - b c + c^{2}$"
      ],
      "text/plain": [
       " 2                2          2\n",
       "a  - a⋅b - a⋅c + b  - b⋅c + c "
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "part(P.Q,[0,4])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "57c709df-f559-438a-a2be-020f510dad36",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle if\\;\\frac{\\left(a + b + c\\right) \\left(a^{2} - a b - a c + b^{2} - b c + c^{2}\\right)}{a b c}=0\\;then\\;a + b + c\\;=\\;0\\;or\\;a^{2} - a b - a c + b^{2} - b c + c^{2}\\;=\\;0\\;$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sE(['if ',P.Q, 'then ',part(P.Q,[0,3]),'=',0,'or',part(P.Q,[0,4]),'=',0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "a8d4f6d9-e7db-4d8f-871c-3d272fb6bb1e",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle a + b + c\\;=\\;0\\;then\\;a\\;=-b-c\\;,\\;b\\;=\\;-a-c,\\;c\\;=\\;-a-b\\;$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sE([ part(P.Q,[0,3]),'=',0,'then a =-b-c , b = -a-c, c = -a-b '])  "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "50d735c8-da5f-4358-85be-b1a0f5794a59",
   "metadata": {
    "tags": []
   },
   "source": [
    " <img src=\"images/p28.png\" alt=\"ml\" style=\"width : 100%;\"/>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "1e701041-0d73-4e9b-80b2-bbc70aee1946",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle M=\\frac{a + b}{c} + \\frac{a + c}{b} + \\frac{b + c}{a}$"
      ],
      "text/plain": [
       "M=(a + b)/c + (a + c)/b + (b + c)/a"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M=symbols('M')\n",
    "P2=polyclass(M, (b+c)/a+(c+a)/b+(a+b)/c);P2.Q"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "6c77d3fe-a3da-4a1c-99e8-a06286a132bd",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle M=-3$"
      ],
      "text/plain": [
       "M=-3"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P2.subsubs([a+b,a+c,b+c],[-c,-b,-a])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "f8d1160c-dca3-4c84-88b0-ed96112bcffa",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle if\\;a^{2} - a b - a c + b^{2} - b c + c^{2}\\;=\\;0\\;then\\;a\\;=\\;b\\;=\\;c\\;$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sE(['if ',part(P.Q,[0,4]),'=',0,'then a = b = c'])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "12dfdbf7-5979-4e52-bc01-b6b289469ce0",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle M=6$"
      ],
      "text/plain": [
       "M=6"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P2.subsubs([a+b,a+c,b+c],[2*c,2*b,2*a])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "29d2d038-6388-4ddc-91ac-c76ff191b48c",
   "metadata": {
    "tags": []
   },
   "source": [
    " <img src=\"images/linep.png\" alt=\"ml\" style=\"width : 100%;\"/>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "56668eed-58be-4548-940d-145fa7d3309c",
   "metadata": {
    "tags": []
   },
   "source": [
    " <img src=\"images/p29.png\" alt=\"ml\" style=\"width : 100%;\"/>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e2e6fc77-56c4-4da2-af03-9e09992c5b8d",
   "metadata": {},
   "source": [
    " <img src=\"images/lines.png\" alt=\"ml\" style=\"width : 100%;\"/>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "851a62e1-e199-4451-bddb-ee13019b5ebc",
   "metadata": {},
   "outputs": [],
   "source": [
    "e1,e2,e3= (a*a-b*c)/(b*c),(b*b-a*c)/(a*c),(c*c-a*b)/(a*b) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "ac167971-3420-425d-90d6-6f22540bb0e9",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{a^{2} - b c}{b c} + \\frac{- a c + b^{2}}{a c} + \\frac{- a b + c^{2}}{a b}=0$"
      ],
      "text/plain": [
       "(a**2 - b*c)/(b*c) + (-a*c + b**2)/(a*c) + (-a*b + c**2)/(a*b)=0"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P=polyclass(e1+e2+e3,0);P.Q"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "8bcef559-71ce-4040-bbe2-1f5ad8183bcb",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{\\left(a + b + c\\right) \\left(a^{2} - a b - a c + b^{2} - b c + c^{2}\\right)}{a b c}=0$"
      ],
      "text/plain": [
       "(a + b + c)*(a**2 - a*b - a*c + b**2 - b*c + c**2)/(a*b*c)=0"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P.opemat('f',1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "5b288664-2de5-45e9-a2c2-0177180e6c80",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle a + b + c$"
      ],
      "text/plain": [
       "a + b + c"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "part(P.Q,[0,3])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "id": "8be51601-297a-45e3-863f-2f977973cbba",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle a^{2} - a b - a c + b^{2} - b c + c^{2}$"
      ],
      "text/plain": [
       " 2                2          2\n",
       "a  - a⋅b - a⋅c + b  - b⋅c + c "
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "part(P.Q,[0,4])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "id": "75916f6a-b8c4-418e-9ec3-26ec6598874f",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle if\\;\\frac{\\left(a + b + c\\right) \\left(a^{2} - a b - a c + b^{2} - b c + c^{2}\\right)}{a b c}=0\\;then\\;a + b + c\\;=\\;0\\;or\\;a^{2} - a b - a c + b^{2} - b c + c^{2}\\;=\\;0\\;$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sE(['if ',P.Q, 'then ',part(P.Q,[0,3]),'=',0,'or',part(P.Q,[0,4]),'=',0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "id": "275d027a-f760-400d-8f6a-8a88feb717e1",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle a + b + c\\;=\\;0\\;then\\;a\\;=-b-c\\;,\\;b\\;=\\;-a-c,\\;c\\;=\\;-a-b\\;$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sE([ part(P.Q,[0,3]),'=',0,'then a =-b-c , b = -a-c, c = -a-b '])  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "id": "a5c17d42-89db-45a9-b2a5-97c317ccbfdc",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle M=\\frac{a + b}{c} + \\frac{a + c}{b} + \\frac{b + c}{a}$"
      ],
      "text/plain": [
       "M=(a + b)/c + (a + c)/b + (b + c)/a"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M=symbols('M')\n",
    "P2=polyclass(M, (b+c)/a+(c+a)/b+(a+b)/c);P2.Q"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "id": "8f6704bf-f1dd-4402-b285-c649931b0db4",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEwAAAAOCAYAAACFB/pMAAAACXBIWXMAAA7EAAAOxAGVKw4bAAACfUlEQVRYCd2X0VHcMBCGDZMCuJSQdBBIB9ABkApydBAmb/cKHUAJgQ64VECgg6SG6+DyfULrkW3dHGQmg83O7Gm1Wsna//6V7J31et2Uslgs5vRP0MPsP8F3W8b0bca/4btAV+gv9GrbHGJGKez7Axs7y5vbo7V/gX+pb6cPmE6FgBuaY/Qc+1JfTRj7hP87auwR/bRwLXbsPvYuQIITgDXYEugKTcTZrSVBkKjeoTLmYy2m8B2EzbzJgpVzEJw5efjnh/zIhqRoqoDhtxxN/g/aAoLdkYy+C0Z8Z3yCnUf2LEnUJOTY2jrePbkHv/sEXqOeR6eDURyMyUIBfY/uoTJy0kJOkmRWJoEv2GZZbmRYzPmNIRg1Oc4PiMth6uU4yJH8zM3L7Az72oABwxiQOQ8OIjKo0YcmO/dFPS1Ae5R90vlZwlr+WwH0s+YQ9Mg8b+//LjzHi8z9fUbNy0pLMgAMr4HBlgDJBQK8xDgWjdou49Oi236Y295C22JfY5z9CVIiALbkeKBNt2QNsHR+uVEnopqyLmSOL71m0OofzfnFftzLz7yn2O+2ViA2Vgdjt6jkuKGd1QDrP8Dg9GrBBNEuX2JllxKMfOq90m9ObP9fH898K6mh7QNoSZrrYQewPCHOL+cqlqJnmP9eQxtlatfza4Wv/wDHNgrxozvD2JP5pdyxZ6hEGUgHMEZFsWSQEwRIf1uKOrPofzG72MzozjAByiAtK2Ad5HyXu5F5br0+SwbpvkdFvwMkcdJXv+NvRc5JpPM+SZ4eQ+YpNqv0LYnhd6MAeIgLmN9T6bUhT7Ak46B3gS9oGW9JfiWmSmPGJiPkYNWUry9i0n58/wWXIO1ulaMy2AAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$\\displaystyle M=-3$"
      ],
      "text/plain": [
       "M=-3"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P2.subsubs([a+b,a+c,b+c],[-c,-b,-a])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "id": "fae0a1d3-0686-400b-8bb9-8f85a71c6e0f",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle if\\;a^{2} - a b - a c + b^{2} - b c + c^{2}\\;=\\;0\\;then\\;a\\;=\\;b\\;=\\;c\\;$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sE(['if ',part(P.Q,[0,4]),'=',0,'then a = b = c'])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "id": "0a4abc97-7663-4620-97df-3f749e6492cc",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle M=6$"
      ],
      "text/plain": [
       "M=6"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P2.subsubs([a+b,a+c,b+c],[2*c,2*b,2*a])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ce0410e5-2ea1-45df-9ef4-bdc480edf407",
   "metadata": {},
   "source": [
    " <img src=\"images/linep.png\" alt=\"ml\" style=\"width : 100%;\"/>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "40ac633b-c179-4dff-82d1-de4109c80295",
   "metadata": {
    "tags": []
   },
   "source": [
    " <img src=\"images/p30.png\" alt=\"ml\" style=\"width : 100%;\"/>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a9915a11-7407-4a3c-aede-0dd55828d182",
   "metadata": {},
   "source": [
    " <img src=\"images/lines.png\" alt=\"ml\" style=\"width : 100%;\"/>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "id": "8bd89f61-4c5a-44c5-825f-a5e00ebd7eab",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{a}{b} + \\frac{b}{a}=1$"
      ],
      "text/plain": [
       "a/b + b/a=1"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a,b,=symbols('a b')\n",
    "P=polyclass(a/b+b/a,1);P.Q"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "id": "4a4e8a04-0f82-4236-954e-783ca26995d8",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{a^{2} + b^{2}}{a b}=1$"
      ],
      "text/plain": [
       "(a**2 + b**2)/(a*b)=1"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P.opemat('f',1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "id": "c983221b-a1eb-44b8-a5f5-f8fcf752a494",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle a^{2} + b^{2}=a b$"
      ],
      "text/plain": [
       "a**2 + b**2=a*b"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P.ope4('M',a*b,1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "id": "ab77d7ea-3030-496d-85c2-183397a86e57",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle a^{2} - a b + b^{2}=0$"
      ],
      "text/plain": [
       "a**2 - a*b + b**2=0"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "exp1=P.ope4('S',-a*b,1);exp1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "id": "885a3356-3253-4610-9554-deca2aecd004",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle a^{3} + b^{3}=0$"
      ],
      "text/plain": [
       "a**3 + b**3=0"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P.ope4('M',a+b,1,'es')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "id": "1ec28f80-3351-47a3-a685-de4da77e49cd",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle if\\;a^{3} + b^{3}=0\\;then..\\;a^{3}\\;=\\;- b^{3}\\;$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sE([ 'if ',P.Q,'then..', a**3,'=',-b**3])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "id": "398a136c-9a43-43c8-9df3-6a471481556d",
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "def f1(a,b):\n",
    "    return(   kpow(2*(a/b)+ kpow(b,3)/kpow(a,3),3))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "id": "b094fe40-2218-4a94-a31c-4d74c6e5219d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left(\\frac{2 a}{b} + \\frac{b^{3}}{a^{3}}\\right)^{3} + \\left(\\frac{a^{3}}{b^{3}} + \\frac{2 b}{a}\\right)^{3}$"
      ],
      "text/plain": [
       "          3             3\n",
       "⎛       3⎞    ⎛ 3      ⎞ \n",
       "⎜2⋅a   b ⎟    ⎜a    2⋅b⎟ \n",
       "⎜─── + ──⎟  + ⎜── + ───⎟ \n",
       "⎜ b     3⎟    ⎜ 3    a ⎟ \n",
       "⎝      a ⎠    ⎝b       ⎠ "
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M=f1(a,b)+f1(b,a);M"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "id": "3a6cd606-512e-4c60-8975-de5d693a5bc5",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAO4AAAA3CAYAAAAL33bFAAAACXBIWXMAAA7EAAAOxAGVKw4bAAALHElEQVR4Ae2d25XUOBCGezgTwDAbwQ4ZwBIBkAEsEQxkwB7eeOOwGQARcMmAJQIWMlg2gh0mg9n/86iM2+27ZLfkVp2jtmxLpdJfVbq67aOrq6tNpngRePHixZmke6hwqXCicFfhpa5/0zHTihHo0v3xiuu9lqq9UkW+S4l/UiEdn+nwWeEm55lWjUCr7rPjxq/3lzURf9H599q1fLpOBFp1nx03coWrh60PiRk20xJnWjkCXbrPjpuI8t0Q+bHEfa34m0TEzmIGQKBJ90dDFqeU8YMzmL8CyHEwLIQbi0lg90hxFpe8qMLvk+LFnNeL4YyZJd9B20xFV7PovtdxnQIwlNzKTzB04XZb2d4q3FM8hPPC76vCHfGrD6N1ef8kuXDag7eZOXXfOVR2CmBF82CcVnVl++UPZ/6/6XjBua6XTqK49aTFfZ3fcul3DuRTYJEBZ2tNt5NRF5SPcv5VwOmtfHN+yrZrisZBkhOn7bQZpenFOI7a+EmBzhRm0f2NNtFU4BPdu62jGXFb0tVcV10xqFc6PnXhjs5xjq86v28VVfxS4YHOWd3tnT4o7UfS6YhRDybKUeILheoqMnJw/b1CVCR5e21GaQZhHFXFPIRRfWfRfaPjOnBfS95HHjKnmJXV2vOq4MKChgtHaXI6hq2fqunb4uLzVPdoCDHuMYQOnivfM4LiNBgMk63nHcNrtrSSB4ccYjNjMZ5N5qUYC5vgum+c46qgf1SpjzoeTG+LElXfHzqcKNxUvHQMxXFatmFuKV70fjrS8+G0W2l13krKAw/mu78qXvJvzZDQDdVnkM0o3WCME6p+r6ihdb/T46oAeoRTHQ/KaR3yDHuZn7U5FU5tVAyVO9JauvKotAybcPxV7cOqXmNsZgzGJXapR0LrvmlxCqM6mMWoqkEI3LapAUPije5XF4PocVl8YPgKsfDESirO2UU0iKRjLl2du3blif3eYJtRncdgvFF6huA2XflPcfB9qOtRb4e1KCyY7rd6XIGBEdKrsBKWSQgIE5zWjKfARNfAyK6/0TlGhFI+KE7aVtJ9ehwaANInT6qPt82Ih2G5hYmu0ziyGk8jx8o+ODOP5rw6+tGl+EkyB9P9luOq6kyimdtexg/DYhIyvzXntEIxKKjcm61gxjy2j2gYn6RofA0VC2EzOxg7bLiOw1ZHJsS7pjMNIkZ1KYjuS8cVONbqwTiTEBAmtO5s42CcVWJ+yzC5bOAUtx6APwH0ES0vNHaF+TpXJL+qs7fNiEcbxgy/wbS+7UWj2TcdiQShRjGC6L50XBXxXIH9yeo8rrHkQ7goHAqn0rHutFSfhx/+ruHANejL9aH9VzxxeBTYxLs9Y3x3vGymB+PfwchhVdRccRyZqcigLbgiU2Q/oXRfdVxaMmsNIqvusuIIXIa7bP2UjqX4GcFJQk/D3KtKtugyFEOGgVWeVV6pxCfbTBfGuoeDEuqdCM680f2hGJM8RvLWfeG4AgJDBKhkW7JQ2nFY3NVxa6FE/HHmi0o55bxLacEOo+KJq8tKmq6oDQFtvtyVNrp7DqdJNuPyDsG4PnqhcSycWTxYWbaGNDp8egTy1v2xK+CxO6bekvXg1X3bGQKtIUM05l5Gp4rw1BOrmhBzrKrRkOe97g/eRlNapiU4OfPlwfmUNhaaZDOqM7j1Yqx0OGiJsc6ZunBuNorjJznXldzeujfHpceFWdmL6PwQCYPCOJoWjarDtnOlYUuCrRD2b/mP7BQjYp4M9inSVJsZivE9gfLWYcyC3zsFGjnDnfOUyUv35rhNiy2zgyKloHwUGcWzt5KHPxX0ktLRU5bz394M7QloDJIcKkvuSTYzEmNbN6gi2HSten/RuIcNe+n+WAUzTyEs0tu68nhe90IB5ZfDIcUPjXi+dyNM7ivYEDB6DJwOF7OZ2ABx9fe1YS/d0+Oa4xSM5gZJlaa3KlpNxRlqeg8VxYdeixXa1OaKDJcgMFjEcQNhtajNFAhF9CMMQ9iwl+5vCA9TwiI97kz40/oTUiPDfNQf7D0rGQKrNdiMJ4ze2b10j+OeOhGMkbdEmcEwBFzLTWLTwbCM+09l8mabmagLX93juNZTXU6UIWfzR8B04M9pGQ4mb7YZf7wNy1GccFx7tpbFokzLI4DxWw+2fOnTSsw2Mw23eq7Juq/2uHWm+XwZBGgwJ7W6y4jXWEpq8jZWIoKLk3XPqnLR2lfG3J31UTqU9llhjPJ4t+y3TsYDbooHTzM17XtaHZr2VvkXT1R7fw1VDd7jzoyV4X3ZUJfoLgmLvdnsADAm6R7HHUUCAWUNelBhFOMBiVV2k2NudJ3niNkOskcSB3DbTqK8QT9bKH5H2yUse6byZ8Nqak1CY2xy9GGt+3uzWZMx9BHHLea2qtyJq2DoMpLg16f8mStR6GDmMkKyn2Qze8Y4ZP1D8pqke+a4NtyZ1GWHrMGB8gJ300EqEJi82Wb8NDZZ9zguL+CCTq4P+XcPCKS2H5ptJpyRTNI9Q2VrPcOJ0sNJQyb+WEBDwbPKEF8K+KbjFx0nz1MLTun9gMOk4dIeq7q4zeyxro1FB7Lhybov57iSbrFhjyod+ypvo7JCXxQOKA5KzRGsoVnMZq5hiufX14Z9dY/jWld9Fg8soyXB8FMzfippmNvQc3TFJ2QIgVVwm3GGzEiMUdiFzpd8fnsCjN5ZvHTP3/rY50SKvWzxeFdfDCT/Iv+sCSFrjYcpzxyhdjv8aQisxCO4zYgnDcoDHXmXl/1zJjwA8XD00v0NVw8Mx+ab8VRt/ZKY8pjfp0Zz2cxtAXEI7z7z0r05LoZjjFIzoJTlvYvw6mUW63EDghXcZoTDfSdfqiOoMfB66d4clxaOBzBOxpSc03ojgKGmaqRz2AzvlEr5KwVjDMJL98euJJtTwOzjmNIPIa0aNEYjvK6VeRgLSWDk9eEpx5OGMtVh4Rw2g/0xf+bNKBALVEM+pFYkTuUnhO6LHleMGPZglPbKzVQwmF1OYYMxsWAS+sNT8IWSbChD24z40Ygxv6WRtG810Vj2fkhNaVIjb93bUJmK85JmY5gaELPI64yJLYo5Pjxlw8IU57eGd0ibMdub+iE1kymFo7fuq45bPM3kepgUKr+EjHN+eIp/NFVfur5EfUKXEdJmMGaGyYz8CnINJ3H74/71jfR/vXVfOq5AYpGE1j8/1fTTMPisSPAPTwlre+F6am+l/ImMYoFthu1ImzdbObZF+cUupH4MpfvScR0g9DBPxJz5xkGTwwAc6nusOPNG931Wg5m7reU7xKFshvktawlVsk7EB+sqvxjiQXS/5bgyRnoAhip8PjHTNQL11h5jKpxZeI3+8JTyMJc7U0CByVNgmynn++JLo0kjOeZDalHjGVL3W47rao1BPXPARQ3EnMKp/jRgWw8Z6Jp9eMqGdHx4qjS2gfJYizs230D2e0kWwmZYXadBM2L+POpDapYx4mMw3e84rgyRXheDzb3uZsOHp3BOGjKGhDgsiyinXNPxncJgUh56W8L54EwJJFS9QtgMmPANKbBm0Y4PqTW+ficBSHZEVF2C6v7o6mr3VUsqhPkGL4QDyDX1DDuALnlBWP5Qeec60rusirLNdKsztO53elyKVyH0uC8VGK5kCoCAMAVLVqhX57TAk22m3Ujm0H2j4zpF8CYKnhtliJjJAwFhyL4db6G0VVIPbvFmVf2yzdTUM5fuWx2X8p2h8SV223esiZVP+xAQdkw7WC9gvrx6yjbzU8Vz6r7TcRFBhbMYwwvNmVxnGoGAMGNL461C+RjfiOzJJs02U/jNrLr/H7yOgQIphgM5AAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$\\displaystyle \\left(-1 + \\frac{2 b}{a}\\right)^{3} + \\left(\\frac{2 a}{b} - 1\\right)^{3}$"
      ],
      "text/plain": [
       "          3            3\n",
       "⎛     2⋅b⎞    ⎛2⋅a    ⎞ \n",
       "⎜-1 + ───⎟  + ⎜─── - 1⎟ \n",
       "⎝      a ⎠    ⎝ b     ⎠ "
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M=M.subs(b**3,-a**3);M"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "id": "db772514-5ace-4d1e-a766-3562f6c2ee3e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle we\\;will\\;make\\;..\\;\\frac{2 a}{b} - 1\\;=\\;x\\;and\\;-1 + \\frac{2 b}{a}\\;=y\\;$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sE(['we will make ..',part(M,[0,0]),'= x and',part(M,[1,0]),'=y'])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "id": "5c9954c0-7370-42c6-b167-2c6f02f658de",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle remember\\;that\\;\\left(x + y\\right)^{3}\\;=\\;\\left(x + y\\right) \\left(x^{2} - x y + y^{2}\\right)\\;$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    " sE([ 'remember that',kpow((x+y),3),'=',(x+y)*(x*x-x*y+y*y)]) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "id": "a0050037-3686-419e-a750-a7e7e7bae5a5",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left(x + y\\right) \\left(x^{2} - x y + y^{2}\\right)$"
      ],
      "text/plain": [
       "        ⎛ 2          2⎞\n",
       "(x + y)⋅⎝x  - x⋅y + y ⎠"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "w1=(x+y)*(x*x-x*y+y*y);w1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "id": "d1e70a2b-cd49-4c59-92c6-96c5f6d6fa8d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{2 a}{b} - 1$"
      ],
      "text/plain": [
       "2⋅a    \n",
       "─── - 1\n",
       " b     "
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "w11=part(M,[0,0]);w11"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "id": "2bf5c96f-c628-426f-89a0-5893a6e26139",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle -1 + \\frac{2 b}{a}$"
      ],
      "text/plain": [
       "     2⋅b\n",
       "-1 + ───\n",
       "      a "
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "w12=part(M,[1,0]);w12"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "id": "9d48824f-6ee6-4859-bc3c-35dd2ca270f1",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle Q=\\left(x + y\\right) \\left(x^{2} - x y + y^{2}\\right)$"
      ],
      "text/plain": [
       "Q=(x + y)*(x**2 - x*y + y**2)"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Q=symbols('Q')\n",
    "M2=polyclass(Q,w1);M2.Q"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "id": "2660b530-1787-4abb-bf54-4ae6e6de3a01",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle remplacement....\\;x\\;=\\;\\frac{2 a}{b} - 1\\;and\\;y=\\;-1 + \\frac{2 b}{a}\\;$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sE(['remplacement.... x =',part(M,[0,0]),'and y=',part(M,[1,0])])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "id": "9f3592d8-b1fa-460f-a2c4-4bd7c4192a41",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle Q=\\left(\\frac{2 a}{b} - 2 + \\frac{2 b}{a}\\right) \\left(\\left(-1 + \\frac{2 b}{a}\\right)^{2} - \\left(-1 + \\frac{2 b}{a}\\right) \\left(\\frac{2 a}{b} - 1\\right) + \\left(\\frac{2 a}{b} - 1\\right)^{2}\\right)$"
      ],
      "text/plain": [
       "Q=(2*a/b - 2 + 2*b/a)*((-1 + 2*b/a)**2 - (-1 + 2*b/a)*(2*a/b - 1) + (2*a/b - 1)**2)"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M2.subsubs([x,y],[w11,w12],1) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "id": "02835951-3614-41ba-91e2-30a0326d6b7d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{2 a}{b} - 2 + \\frac{2 b}{a}$"
      ],
      "text/plain": [
       "2⋅a       2⋅b\n",
       "─── - 2 + ───\n",
       " b         a "
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# get the first monomie\n",
    "w3=part(M2.Q,[1,0]);w3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "id": "b1d27f13-1646-4095-aab0-7462c5d14abc",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{2 \\left(a^{2} - a b + b^{2}\\right)}{a b}$"
      ],
      "text/plain": [
       "  ⎛ 2          2⎞\n",
       "2⋅⎝a  - a⋅b + b ⎠\n",
       "─────────────────\n",
       "       a⋅b       "
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "w3=factor(w3);w3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "id": "a454ab8e-1da1-4016-b813-501ceb753411",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle if\\;a^{2} - a b + b^{2}=0\\;then..\\;\\frac{2 \\left(a^{2} - a b + b^{2}\\right)}{a b}\\;=\\;0\\;also..\\;$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left(\\frac{2 a}{b} - 2 + \\frac{2 b}{a}\\right) \\left(\\left(-1 + \\frac{2 b}{a}\\right)^{2} - \\left(-1 + \\frac{2 b}{a}\\right) \\left(\\frac{2 a}{b} - 1\\right) + \\left(\\frac{2 a}{b} - 1\\right)^{2}\\right)\\;=0\\;$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle finally..\\;\\left(\\frac{2 a}{b} + \\frac{b^{3}}{a^{3}}\\right)^{3} + \\left(\\frac{a^{3}}{b^{3}} + \\frac{2 b}{a}\\right)^{3}\\;=0\\;$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sE(['if  ',exp1,'then..', w3,'= 0 also..'])\n",
    "sE([M2.q2,'=0'])\n",
    "sE(['finally.. ',f1(a,b)+f1(b,a),'=0'])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "94016870-d1d0-4f55-8790-e617bad60177",
   "metadata": {},
   "source": [
    " <img src=\"images/linep.png\" alt=\"ml\" style=\"width : 100%;\"/>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "73391682-6b29-4464-ac69-c0816d51ad0d",
   "metadata": {
    "tags": []
   },
   "source": [
    " <img src=\"images/p31.png\" alt=\"ml\" style=\"width : 100%;\"/>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "06cf9b86-63e9-4097-bc2f-0fd271eca1d4",
   "metadata": {},
   "source": [
    " <img src=\"images/lines.png\" alt=\"ml\" style=\"width : 100%;\"/>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "id": "de636695-25e8-4dd2-8ee0-2e2fe782c796",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{1}{y} - \\frac{1}{x}=2$"
      ],
      "text/plain": [
       "1/y - 1/x=2"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x,y,=symbols('x y')\n",
    "P1=polyclass(1/y-1/x,2);P1.Q"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "id": "0a9dad8e-685a-4a17-baf2-a3e604b90294",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{x - y}{x y}=2$"
      ],
      "text/plain": [
       "(x - y)/(x*y)=2"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P1.opemat('f',1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "id": "b2a012da-17e1-4a8b-8724-92311a3b0828",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle x - y=2 x y$"
      ],
      "text/plain": [
       "x - y=2*x*y"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P1.ope4('M',x*y,1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "id": "bd1cc58e-708d-4ffa-a49e-67c1423714a1",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAMsAAAAXCAYAAAClB7tnAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAE9UlEQVR4Ae2a3VEcMRCED+oCwA4BMsAmA8gA7AiADOzijTcKMgBCwBkYRwAmA5yB8WWA+1sklW6NYPV3aOusKiGtTjvq6ZE0Iy0rj4+Pk9h0fHy8pneOzHvrptxX+yxW1qL6jxFzKjfLpGsqR6H3XuJuGnrplfZTCT20fVQ/V/2n8oZta7AcI+ZUGpdJ11SOQu8FuVsNvfFK+4EWyLbX51T1dbVtem2tVceIOZXDZdI1laPQe0HuUhcLXuU2NFqj7WPEnErlMumaylHovSB3Kylnlv4o8ih4ll2VLYdhc7DHiHlOgYiHZdI1gpZBXX3uUs8sbiAJI/TaVf7gGhuvtIRZWLgg+Woo+6jygWe135m2rKIlXYcoUpuPIRhsnz53WZ7FKMbhfk/1Zm/CrPKULWE2WDhQ7lmMquOlvyjvqH5t21NKI3809jF4q/ERw+Fz3CV7Fk+xHUDoGQ8zU/krBtQi+zaImYWx73MgjHiVA7VdKb/zf4upv6WuGpvLHy58LmIwq281PmJwhLhLOuAbYexYJ6pvklXnYEQI0WRqFDOT6o+w8d3KT3iUNYPZbx9Ub0BX9OnrNAR7FT6GDGz7vMRdqmfhmwpkULqkgVgwraYWMbMo2GxmAdJSJhyiWtQ1oOJccy0+5gZ55SHInTuzyGAYBvdP2lImPODw+VmZdKM+356q7fz1cHMTd6/nM4vO/Hap0p0J7G8tl8J7L3yEMSsWp9GlqH08mUW5k1wufMDvbGH1SCklJ4uPUnr6nsV9uZRwDpiXyiwQYmhCLuLJ5haLMB0ZjBgIzL6BPumZ9uxkOCBMiEl3ei9qoao/Ia1/Q2bHq2GfhXBnFUgpC/FRRM9usQgQOxaLwU9MMnul+V71/u9+3zepCzeT6sYMzkVD/3KBtqwbJSN7orEWFWJysL/QeG7Rq17cPpK5MO4sh4llFh8l9bSe5VZC/YnGJGNX7NpURu2OIVIkh1Dvh3JMLM61dOibw4N+s94OL3LSGxtP0G/rdWnnUbrgwa9V9hdmDftkc2fwPudt2Vwnz+hB82Bva+Tn8pGtJ6BJ1rP0JyMEWK/SdSzxR8rPJKfYx0sjb6ISvCxAd1WpNsIZ2op4FsmpmoS3O4+o7C+UidqK20cysQWyk7nTu/9gNTKzzyySXYQPycnW0xq+Wyz2wSgKeSS7Y080IJOOshuYemMJz8eO5eNDD7779CdaEnTJYde33AyVMWgXlWwm14ZKN/lUJ0yaqPQ9Ps8WQyn7VOduKFm2n3SswUe2nlMBYyEQF3KAZBfuDKa6b6TugGSVabBkYvl4gVjsvIIw8eEmMs+lkuTiAbdU9j05E4azS237VOcuhquKfGTrOZUi7FRkbr0wzIOy26HVxm/2EK1qk4mF0sXJoDOYwV1lgjNGiSScGJCNirgcz2UTuvD95UyZRVPTPs1wV5mPbD2nMgTehFgfg0wE+FD5uzLG4wONf0CiS4uJXZnvKdzY/Va2//3c+nmFhcKC6eJzlX6y4WNt+7TEXU0+svV0HyV9K429bhbNtspilwlj52Qo/hLcSQbesNhHyaHYY/ql6LkaM0CLfaX0uTIesEuqE0qyU588tfz/G2KgIneE8S6UD42/qPZSehKGjT0RPvrxPl/xORi726KxK1gRfxXuxH1r4W8RPUcfhskwncs3E4qzylWDxqo439NFLwt3pfT8C1eWKQbpiFB1AAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$\\displaystyle x^{2} - y^{2}=2 x^{2} y + 2 x y^{2}$"
      ],
      "text/plain": [
       "x**2 - y**2=2*x**2*y + 2*x*y**2"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P2=polyclass(P1.ope4('M',(x+y),'e'));P2.Q"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c19abe16-cb53-4d9c-819e-8d26038b0dc5",
   "metadata": {
    "tags": []
   },
   "source": [
    " <img src=\"images/p31.png\" alt=\"ml\" style=\"width : 100%;\"/>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "id": "c03fb590-1663-46a2-a65b-ebe54b47b9a6",
   "metadata": {},
   "outputs": [],
   "source": [
    "def fxy(n1,n2):\n",
    "    return(kpow(x,n1)*kpow(y,n2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "id": "c2b9dce2-328a-4ac8-8374-7c3447d4bf50",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{16 x^{4} y^{4} + \\left(2 x^{2} y + 2 x y^{2}\\right)^{2}}{x^{4} y^{2} + x^{2} y^{4}}$"
      ],
      "text/plain": [
       "                            2\n",
       "    4  4   ⎛   2          2⎞ \n",
       "16⋅x ⋅y  + ⎝2⋅x ⋅y + 2⋅x⋅y ⎠ \n",
       "─────────────────────────────\n",
       "         4  2    2  4        \n",
       "        x ⋅y  + x ⋅y         "
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(16*fxy(4,4)+kpow(P2.q2,2))/(fxy(4,2)+fxy(2,4))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "id": "c5314f68-ce1c-426f-858c-1291ddbbfad0",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{4 \\left(4 x^{2} y^{2} + \\left(x + y\\right)^{2}\\right)}{x^{2} + y^{2}}$"
      ],
      "text/plain": [
       "  ⎛   2  2          2⎞\n",
       "4⋅⎝4⋅x ⋅y  + (x + y) ⎠\n",
       "──────────────────────\n",
       "        2    2        \n",
       "       x  + y         "
      ]
     },
     "execution_count": 59,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "kres=simplify((16*fxy(4,4)+kpow(P2.q2,2))/(fxy(4,2)+fxy(2,4)));kres"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "id": "074176c6-d5ec-47de-a6db-04809ceee6b9",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left(a^{3} + b^{3}\\right)^{2}=0$"
      ],
      "text/plain": [
       "(a**3 + b**3)**2=0"
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P3=polyclass(P.ope4('P',2));P3.Q"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "id": "c1d131b0-66e7-4920-9159-7dfee84e1680",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle a^{6} + 2 a^{3} b^{3} + b^{6}=0$"
      ],
      "text/plain": [
       "a**6 + 2*a**3*b**3 + b**6=0"
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P3.opemat('e',1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "id": "706900fd-829f-4824-ae06-b1e10d1aeb45",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle a^{6} + 2 a^{3} b^{3} + b^{6} + 2 x y=2 x y$"
      ],
      "text/plain": [
       "a**6 + 2*a**3*b**3 + b**6 + 2*x*y=2*x*y"
      ]
     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P3.ope4('S',+2*x*y,1,'f')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "id": "86519d45-0b7b-422d-b50b-1b3ce61d66ac",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle a^{6} + 2 a^{3} b^{3} + b^{6} + 2 x y$"
      ],
      "text/plain": [
       " 6      3  3    6        \n",
       "a  + 2⋅a ⋅b  + b  + 2⋅x⋅y"
      ]
     },
     "execution_count": 63,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r1=P3.q1;r1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "id": "b392d166-89d5-4285-8d16-7bcc0002a0c0",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 2 x y$"
      ],
      "text/plain": [
       "2⋅x⋅y"
      ]
     },
     "execution_count": 64,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r2=expand(P3.q2);r2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "id": "022deff9-8ef3-496d-8405-6065fe45cdcf",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle a^{6} + 2 a^{3} b^{3} + b^{6} + 4 x y=4 x y$"
      ],
      "text/plain": [
       "a**6 + 2*a**3*b**3 + b**6 + 4*x*y=4*x*y"
      ]
     },
     "execution_count": 65,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P3.ope4('S',+2*x*y,1,'f')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "id": "1d59feb1-b589-4c9f-93a1-8bfd3b91762e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle a^{6} + 2 a^{3} b^{3} + b^{6} + 4 x y$"
      ],
      "text/plain": [
       " 6      3  3    6        \n",
       "a  + 2⋅a ⋅b  + b  + 4⋅x⋅y"
      ]
     },
     "execution_count": 66,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r3=P3.q1;r3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "id": "c348c7f7-d2b6-4d32-b445-0bbc0b8be2cd",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 4 x y$"
      ],
      "text/plain": [
       "4⋅x⋅y"
      ]
     },
     "execution_count": 67,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r4=expand(P3.q2);r4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "id": "71f2b318-0a62-4e9d-a0d6-6adf0ee44186",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{4 \\left(4 x^{2} y^{2} + \\left(x + y\\right)^{2}\\right)}{x^{2} + y^{2}}$"
      ],
      "text/plain": [
       "  ⎛   2  2          2⎞\n",
       "4⋅⎝4⋅x ⋅y  + (x + y) ⎠\n",
       "──────────────────────\n",
       "        2    2        \n",
       "       x  + y         "
      ]
     },
     "execution_count": 68,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "kres "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "id": "67dd1886-3491-418b-9db9-4dc91d83f841",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{4 \\left(4 x^{2} y^{2} + \\left(x + y\\right)^{2}\\right)}{x^{2} + y^{2}}$"
      ],
      "text/plain": [
       "  ⎛   2  2          2⎞\n",
       "4⋅⎝4⋅x ⋅y  + (x + y) ⎠\n",
       "──────────────────────\n",
       "        2    2        \n",
       "       x  + y         "
      ]
     },
     "execution_count": 69,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "kres=kres.subs(r1,r2);kres"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "id": "4840fa11-ee5f-4ad3-a49e-b4d082f4df0a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{4 \\left(4 x^{2} y^{2} + \\left(x + y\\right)^{2}\\right)}{x^{2} + y^{2}}$"
      ],
      "text/plain": [
       "  ⎛   2  2          2⎞\n",
       "4⋅⎝4⋅x ⋅y  + (x + y) ⎠\n",
       "──────────────────────\n",
       "        2    2        \n",
       "       x  + y         "
      ]
     },
     "execution_count": 70,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "kres=kres.subs(r3,r4);kres"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "id": "517fb83a-cc83-46d1-9b3d-512bf7b7afda",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{4 \\left(4 x^{2} y^{2} + \\left(x + y\\right)^{2}\\right)}{x^{2} + y^{2}}$"
      ],
      "text/plain": [
       "  ⎛   2  2          2⎞\n",
       "4⋅⎝4⋅x ⋅y  + (x + y) ⎠\n",
       "──────────────────────\n",
       "        2    2        \n",
       "       x  + y         "
      ]
     },
     "execution_count": 71,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simplify(kres)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "795d2551-f65c-4001-a5bc-3d20c4b9f470",
   "metadata": {
    "tags": []
   },
   "source": [
    " <img src=\"images/p31.png\" alt=\"ml\" style=\"width : 100%;\"/>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "id": "a212ccd4-3f47-4c5d-828d-93a854c2319b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{16 x^{4} y^{4} + \\left(x^{2} - y^{2}\\right)^{2}}{x^{4} y^{2} + x^{2} y^{4}}\\;=\\;8\\;$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sE([(16*fxy(4,4)+kpow(x*x-y*y,2))/(fxy(4,2)+fxy(2,4)),'=',8])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "adafe16e-04f4-4131-afba-92098e0b18d0",
   "metadata": {},
   "source": [
    " <img src=\"images/linep.png\" alt=\"ml\" style=\"width : 100%;\"/>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "51070cd0-cf35-4244-b3f1-e6f2325b62f7",
   "metadata": {},
   "source": [
    " <img src=\"images/p33.png\" alt=\"ml\" style=\"width : 100%;\"/>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "127ed665-e77d-40a8-aee4-fb2012ab9821",
   "metadata": {},
   "source": [
    " <img src=\"images/lines.png\" alt=\"ml\" style=\"width : 100%;\"/>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "id": "ac4593e4-fa6c-48c0-9329-2c0ad28766e1",
   "metadata": {},
   "outputs": [],
   "source": [
    "x,y,z,k,w=symbols('x y z k w')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "id": "efe88d3b-01d7-407f-a758-f7c7bd6a0a4f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAFQAAAAQCAYAAACIoli7AAAACXBIWXMAAA7EAAAOxAGVKw4bAAACpklEQVRYCc2X7VEUQRCGF+sCQM0AM5AyAo8MOIlAyUCKf/fPggw4QsAMMAOUDDQDKTI4n2ecGZZxXdi7m4Wu6u356O1+r7e7Z25ruVw2tWg+n0+xvYNc1PIx1G5tTC+GAhqov42+/JyoKqbaAX1OgRwFyyR5oRT8cp/i/B3yI7wDH8S1K3S+xnF10cLzBmc/mZ8mp3HvHDlLa2NK/L7F30X0OWN+DRuraTtDT1g8ldm4gs9VYH6ENNgn8Jh0HLFc4vS4cPyB+X6xNubU2JzBJphx+4w8Qi5ChjIwM8uACdhgSq/gcj9s1HiAx6/tR5X24F9hdPdw7dvddLxRxPYeeZu8Mjaoh85TyX9noQ1awKZxWEP2lhb7fi1P9JL8EA37wVmxqf3/2b1hL7UXs/FL8a6+yrV7Kry/aUzBPnZznBinyk2J12x1XZtQ9C5lCue+dQ/tIye8b5Z7bVrJDu8ZOEv+JeOQEUj71w94l/E1chDxzlqYkjPsWEWHyBxM99o9NOiikDItZUjD2rYcFMZ9mMFmci4v5uK7ZW1wMDcFHd9i2EfmYDIO8ZnEgSeWfcC+FMqTcU5t1jwg8svMxyKzoI1Dv0/WP3VOHAxmV9V5UM0mPFSQLXGjfAPnjIgG0gHB1qhkMEMf1mvEItaunqxKVcK/7cbEsmpsRfJr2DYSzgMDalb611CggrYvXMI2dXtV+4BQZUwSvPdNbxi/Ye+k0pOc8Pg9AIsV0iANolnpRzdmoQV1HkoobISi067yWMk+9gzsFLm7kgFe2jSmEsc/h1KpsObc1pHbxxBb/PAz2AoJxNh25H2597r0V7v3uTKmXqtx05KvRgRhndK0Bdl2ElleC2zm20faGCLXxPSgq6ol/6D3HgV+eLgvRhV750XtYPTAefTWH5Ki+6KoDjITAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$\\displaystyle x + y + z$"
      ],
      "text/plain": [
       "x + y + z"
      ]
     },
     "execution_count": 74,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e1=x+y+z;e1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "id": "2024876c-52d2-4ffb-afe0-21421812c1e0",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle x + y + z=1$"
      ],
      "text/plain": [
       "x + y + z=1"
      ]
     },
     "execution_count": 75,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P=polyclass(e1,1);P.Q"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "id": "7839827f-5031-4bf3-b9b3-1d11e4929829",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle if\\;x + y + z\\;=\\;0\\;then..\\;x^{3} + y^{3} + z^{3}\\;=\\;3 x y z\\;$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sE(['if ',x+y+z,'= 0 then.. ',x**3+y**3+z**3,'=',3*x*y*z])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "id": "0fb85e29-2e34-4b86-895f-7beedbbc52e3",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle x + y + z - 1=0$"
      ],
      "text/plain": [
       "x + y + z - 1=0"
      ]
     },
     "execution_count": 77,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P.ope4('S',-1,1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "id": "b704c6e8-4397-48bf-85f9-0cb0335de687",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle x^{3} + y^{3} + \\left(z - 1\\right)^{3}=3 x y \\left(z - 1\\right)$"
      ],
      "text/plain": [
       "x**3 + y**3 + (z - 1)**3=3*x*y*(z - 1)"
      ]
     },
     "execution_count": 78,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# we will usee z-1  \n",
    "P1=polyclass(x**3+y**3+(z-1)**3,3*x*y*(z-1));P1.Q"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "id": "8347255d-6103-464d-a0e5-b264cff4b961",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXYAAAAYCAYAAADu19b7AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAId0lEQVR4Ae2d7XEUORCGF8oBAJcBzoCPCM5kwEcEQAZH8Y9/Li4DIAIOMgAiwJABZIDPGXDvo1KrtFPjWUkzsyv51FVCGo2kebvVarU+1lz7/fv3JpdevXp1W3UeKlwo3FC4r3Cq/O+Kq6QWMSPIVnGnKIF4Q3de+rLoFPRU+ehVpy6BLoFCCRwV1nutej81AP+mvuK/FH1RuMlzpdQiZkTZKu4UNXgt3XluBZV+o/Q3hWPL63GXQJdAvgSu51dxNU717/uo7h9K/4yea0y2iBk5too7RQeeyZifRAWZxG4r706U15NdAl0CmRIo8tg18IZbLmzLMCirpRYxI8xWcScqAt76WWLZXqxLYHEJaHzhSNTulE7yPcZDkWG3r6hBtmCeKLxR+q3l1xy3iBl5top7ShdGdAZDzxbf0HGYaqa/6xIokoAfUxj1pg278DM5PVRwW+MI41rJ4SkVjdQYB2AfFD7FDdv7GuMWMSPHQ+PW9zngxPhC9DvP7JN/JmMOqQ22X9Cju0of5PB0Tf7myCa1bg0yjLHWLE9hY5fhvuIXMeZa07v6Vu9xsnGKPsLDbMNOI/6jHHoxKJvwtlrEfEhZS14Y8uFh5zPlceD5SO+dQoExl1SXCcLaOZRRX42/XHnklPf98k51zhXuKTBB3lT+QeSobzvyuFbRF/tGaeyxfVF8t7SNfdTzOJP7VuWxwX8qvsg+POVjCv8qxAdcpkQoVjaprRMFjMQqpLabw4wg1sA9Q8D0D4edeDpG//iEXVm0/ORY7ZnX/0DpC4U7Pi+5DSuoenP0aBX+DNtasZcZEysrqfhCw1qfTG23ZnlyHogjUTUV9C08ubPOEsOOEcc7iPeluNlAvg10JbPohkoTViEEpIabwowgVsJdKmNWYsiR4Mjjs8fsWPUx6igjv4HAoOMsYKDoqxJChwgltDh/JSCuUJ2a5flYutbEmWCOPnie4O1G6eHpI33wpRr45T/MD5QOtjeayHyLmGGtCtzqa/bRt36noDzz3ku9H5aOGGLiQGrX9vFD3tqJlfhbG3a17dcqT+HCCY2d0mplWAgM3h4Hwy6GGWC2HYKhfqqAR8WtF+iryrh9VMXMxoSDU4SbH7X80HM4Gfbv3ilmqVoNZhOaMNmBIVkOo/KQOVsKzqOoETdghYsBwrLvuWEl30h5KbxtTRRWt4Z4Cf5q4KMWDFPy1Lsc22NlJ8f7BN84SpOH/Sm6O9H+oV/B24Ng2PUQDjrEGCesbNpjzF8o2N5N8QGZ2lqLWDmAEe8RzMGwK/1YwbxKJasj8CJbfuCF/D8pPla8d49V300iYcNgY9SZ/JkszxTGqDneYOKq8zfsKPGL/tGfOfRd9TCQOylRnjm2Z+54vyfQu1aYTequ74wfik+cYZfw8dTdprt/SYRBtKtAt5QevqfMQUm48W6/ehAPFA+XWORNzs6+7t4jj92dYNvHlRcU3PJqi4UxrHyURke+Kd66FaNn+qU53pC1sF9p/uAxJvG7qhOxS556n2x7VHaJ8Y7Hfx7LIE77bzSpu54PeAt77GdiKDaKGERmZZenOGl2jgU0llY7l3kHTBwbvR9Tsinv4Fx1bBWBd346+C6eyDBvUGT6cQXM7oNqN8hbaZSNidMm0mlQCW99m/z9HtpOJduySiqvb3xUuFDhD4rDFTul1+ZtaT0a5XcN/tQm/bFqv4wyU0HmJfLMsT1LjHdsDTo7SqW6W1G/MvZum8eOlxITBnExI2MNi/kxw71RPp4fv56Kt1Gs2qWxyrsOUgxeBkw46VYeWwbkzfLY1c6imIVni9Q+Xgj71KPf2Sqc8aD2kM1i93TVHvLcKB7qClsxyJ9gk6ySruxavI3KStiK9Mhj3Qt/wrhovzhBV/iP+EyS54g+XWp7vOw2ilcb7yZKfSNLdz22xcab4SiIb6nOxdGwohca2WGQKg8DufHgSdZGrCjw7Bk0RnQ+96KHhsjeHzwWNjByzS9Mokpz5z7mowac9L+7uSJswTOfAqZyTfAGD8J6pfnb0U+XrX6mqk2toovk6fsBnYF22Z454x2Plv6+lFrS3REm4O38yCs1P+Vmfxfv1nlDSofltPLcgYXiWonZNcYLzmr31wEn+aLEY6sUDm4W2friO0uQsDJBXqitzz6Om73nH8LKqCXewA5Pnq8ryV/cWcO0+B5d/QzL5TynylPlMEIltmfOeD/XN6k/SsLUzLgcZWCzwWP/eaR/YITAzRJn7ZVmEDvyjNoBpWXXFmPUYciRxwxPiyut/8SsSPhYpuKl4/lwE4bAzRi2Eqoy6sJjFFYVliHs4EVn2EpyOtMob7B01fmDx33STnl6/SmxPXPGOyv4+2OCaFh3Y3bYDvp0pH/wtNibRsAbMccg5Q96sURj+R0fWFCkRkKJuK/OAeQvBfuPGoIXWRnoJ8LKimKjGOOIl46yIvsqt46E663CiQJ6YYTnw58CiOXcHG8wIx6a5U/Y8XqZYG31xE0l9IjrylnnVqqzCCXKs9T2zBnvOFGxDsf8Vqe7BX2LHX+9yB8BiyVTkhZ4jNvYtkRJcxu1h4HHCK12mLE05iJGe6UtCfQ+2RLH/+Yhd7yrPHe9s26AtSBM8YWjhVN+fL0SwCzj3VI+F4+YeKMQfpKuNJ4Ld2NPc9vKLF+MOfM7vXi6BHqfpMuqyZILjXccvyq3aWd2CisZeNuwFXNwUmfFS/lcPCw94qUV2xosq8PJem6DKeVnYk75RC+TKYHeJ5kCa7P47PEuPcE+4NmySzC8dNGkVOBFwOHHTVhVGPaZkmSWgiH+DAJ763jwcyaKmXB69S6BLoEVJbDUeOeSAmcT7qxrRbz7ahrnNly8qGKPfV+c9+90CXQJdAmYBLyXy1lc+GGjvWspFn62nrmqG1Yf/wGcAwE0BAjSYAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$\\displaystyle x^{3} + y^{3} + z^{3} - 3 z^{2} + 3 z - 1=3 x y \\left(z - 1\\right)$"
      ],
      "text/plain": [
       "x**3 + y**3 + z**3 - 3*z**2 + 3*z - 1=3*x*y*(z - 1)"
      ]
     },
     "execution_count": 79,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P1=polyclass(opemat(P1.q1,'e'),3*x*y*(z-1));P1.Q"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "id": "d9d6478f-7ad1-48c7-afd9-ceeec182a17c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - 3 z^{2} + 3 z + 3=3 x y \\left(z - 1\\right)$"
      ],
      "text/plain": [
       "-3*z**2 + 3*z + 3=3*x*y*(z - 1)"
      ]
     },
     "execution_count": 80,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P1.ope1('S',-x**3-y**3-z**3+4,1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "id": "a66d0214-9a2c-4e2a-988e-7973b94f4683",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 3=3 x y \\left(z - 1\\right) + 3 z^{2} - 3 z$"
      ],
      "text/plain": [
       "3=3*x*y*(z - 1) + 3*z**2 - 3*z"
      ]
     },
     "execution_count": 81,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P1.ope4('S',-P1.q1+3,1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "id": "f1265275-6df0-4ba6-b050-8fc567e31f0a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 3 \\left(z - 1\\right) \\left(x y + z\\right)$"
      ],
      "text/plain": [
       "3⋅(z - 1)⋅(x⋅y + z)"
      ]
     },
     "execution_count": 82,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "factor(P1.q2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "id": "6e1fab22-de0f-4796-bf8b-9d4db4f48d0e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 3=3 \\left(z - 1\\right) \\left(x y + z\\right)$"
      ],
      "text/plain": [
       "3=3*(z - 1)*(x*y + z)"
      ]
     },
     "execution_count": 83,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P2=polyclass(3,factor(P1.q2));P2.Q"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "id": "7c75d1bb-310a-49ed-9fc0-619c6151005c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 1=\\left(z - 1\\right) \\left(x y + z\\right)$"
      ],
      "text/plain": [
       "1=(z - 1)*(x*y + z)"
      ]
     },
     "execution_count": 84,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P2.ope4('D',3,1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "id": "97113c3b-1857-44f7-a43e-36e4336bfa1b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{1}{x y + z}=z - 1$"
      ],
      "text/plain": [
       "1/(x*y + z)=z - 1"
      ]
     },
     "execution_count": 85,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P2.ope4('D',x*y+z,1)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ee46d358-7133-4f51-a76e-5ba0298cd532",
   "metadata": {},
   "source": [
    " <img src=\"images/p33.png\" alt=\"ml\" style=\"width : 100%;\"/>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "id": "5e1d7763-97b8-49a2-919e-ea3d4ae51d49",
   "metadata": {},
   "outputs": [],
   "source": [
    "a1,a2,a3=1/(x+y*z),1/(y+x*z),1/(z+x*y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "id": "8f8ff794-7aec-4544-a1df-f049fc9c916a",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle if\\;\\frac{1}{x y + z}=z - 1\\;then\\;\\frac{1}{x + y z}\\;=\\;x - 1\\;and\\;\\frac{1}{x z + y}\\;=\\;y - 1\\;$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "show_exp(['if',P2.Q,'then',a1,'=',x-1,'and',a2,'=',y-1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "id": "adab340b-c09d-44eb-ba4a-12cda5a72347",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{1}{x z + y} + \\frac{1}{x y + z} + \\frac{1}{x + y z}=x + y + z - 3$"
      ],
      "text/plain": [
       "1/(x*z + y) + 1/(x*y + z) + 1/(x + y*z)=x + y + z - 3"
      ]
     },
     "execution_count": 88,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P3=polyclass(a1+a2+a3,x-1+y-1+z-1);P3.Q"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "id": "fc7b0efd-0783-4cde-a474-e77d78463734",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle if\\;x + y + z\\;=\\;1\\;then...\\;$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "show_exp(['if',x+y+z,'=',1,'then...'])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "id": "e172060e-271d-4dfd-ad88-00c202a35d57",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{1}{x z + y} + \\frac{1}{x y + z} + \\frac{1}{x + y z}=-2$"
      ],
      "text/plain": [
       "1/(x*z + y) + 1/(x*y + z) + 1/(x + y*z)=-2"
      ]
     },
     "execution_count": 90,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P3=polyclass(a1+a2+a3,-2);P3.Q"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "762eba06-02b9-4cd0-8993-a8c4e01fa3eb",
   "metadata": {},
   "source": [
    " <img src=\"images/linep.png\" alt=\"ml\" style=\"width : 100%;\"/>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8b4ecb13-89a9-4592-980f-6e0a4ffa9960",
   "metadata": {
    "tags": []
   },
   "source": [
    " <img src=\"images/p34.png\" alt=\"ml\" style=\"width : 100%;\"/>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0468cfd6-4669-4393-862a-a72ddc4b9151",
   "metadata": {},
   "source": [
    " <img src=\"images/lines.png\" alt=\"ml\" style=\"width : 100%;\"/>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "id": "d50b8324-2913-4541-9e65-c6f0307451c0",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{a}{a b + a + 1}$"
      ],
      "text/plain": [
       "     a     \n",
       "───────────\n",
       "a⋅b + a + 1"
      ]
     },
     "execution_count": 91,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a,b,c=symbols('a b c')\n",
    "e1= a/(a*b+a+1);e1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 92,
   "id": "89fa8345-8223-430f-b059-d6aa87f95d17",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{b}{b c + b + 1}$"
      ],
      "text/plain": [
       "     b     \n",
       "───────────\n",
       "b⋅c + b + 1"
      ]
     },
     "execution_count": 92,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e2= b/(b*c+b+1);e2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 93,
   "id": "61d670a7-52cf-4319-a4de-9c3c7dad8d21",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{c}{a c + c + 1}$"
      ],
      "text/plain": [
       "     c     \n",
       "───────────\n",
       "a⋅c + c + 1"
      ]
     },
     "execution_count": 93,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e3=c/(a*c+c+1);e3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 94,
   "id": "6f822d02-ebd1-440b-8bdf-dd595cc34a03",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAUwAAAAtCAYAAADFsVIUAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAH40lEQVR4Ae1d4VXcOBhc8iggXDogHVwuFRA6yKWDCx2En+FfXtJBkgoCdAAdcEkHRwfH0QE3Y6R9WnlZy15b/j4Yvae1LNvSaMb6LMlaa+fu7m4hJwYsMHBycvIcOM7g/4C/wf5LC7iEQQxEBp7FgLZiYG4GYCBv4Q+B4xr+cm48yl8M5AzIYOaMaN8CA78DxIUFIMIgBlIGZDBTNhSenQG0MN8EEGphzq6GAOQMyGDmjGh/bgaaLjm753MDUf5iIGdgx9tLH1SkfRTiGJ4V6l/4c/i3iP+CrZxzBqDjTxSBY5hXoSh88XOBeOos90gYgJ6fURTWX7oX8D8Q96vZM/zjymCCUHbX+Bb1FcKsVAtsOdbF+D2E1SohKU4d9ONb8v/gWXEOqGcS9xLhRnOnxRNsMAAN2eBhnT1GuHkIYvsV+/vYsndh2rnpkoNMViYaSxKdVhyGrxEnY2n6VisCF8cvG2PJKxJd3xaloJOsM0BjeQ5d0x7DO8S5eMm3a53dBB+b8DSap0kcg6xkKfnZYe06YoAtjF+JkaTBpOZ07LbJOWYAWjYtSRThU1oMxO+l+5bDblqYIJFPoUuQu2xJhsoUm/iWeRa2MgY4Yf3v7FTG0cUxzfs9/XpkoFWHvRXChcEMhpEtjXxQmAIscFxTULzdeevxcv4lX/qk7s+wI41TVpyFN9RhVyVxYTATRvNWBitTY0QhCN+Us7Up55uB5fh0qGR8KB4hvOxZ+C7e00Sf6PfPOgZwPI5frztsJs6FwQxk0zAuDSLi3of92IV7jbhlZTPDsID0YYBj0UuNEeZLvlPo+q1PIjrXLAPU8VWODvry/cRNHm9x3820IpDKLvl3eLYym3lb2LLVQbIZx/HNvMuOaDkvDASNqSdbIZp/6UW4HjiDceQVcQ4mw98Q76IH4cZgklU5MSAGxMCcDLjoks9JkPIWA2JADEQGZDAjEw626La8gefYrdwjYkC6+hFTBtOPVkTKcVx6ucfFgHR1oqcMphOhBFMMiIH5GZDBnF8DIRADYsAJAzKYToQSTDEgBuZnQAZzfg2EQAyIAScMaB6mQaHw1pRfdVn3V7HfAtx1/4rgV37i/64NlkqQpKv/e2Dn48ePWme3oo6oNDtDs8O1/CYkP7Q66OvyuE5aDyW/4Lqh2krXAnKNnLI7VGQj+AWjBwPSugdZjk6VrvXE0hhmPa6VkxgQA84ZkMF0LqDgiwExUI8BGcx6XCsnMSAGnDMgg+lcQMEXA2KgHgO9F0HDADP/98oPu3KtlRvs87uFchkD4IXLLZAnLgl8mx0eust0tkrLk34WsVrVdegNpevaDGzSeIjBZIU9RKJceyV+7byd6xOMCRWcHznmPEk+UPbHpAHpb72uDdJwo58VrB50HfM+e4pplWrc22AmZLIFtbJcZnKsOAignKDNuYUmliHYBg+upTFqJo8j/AFhcmTVjaJfV+G24TNJexSsQ7E40zWhrU5wKK9jo9sGR6nGg8YwAzCWd+sWD9J4HjzTs+Cs4Rmdk5H168K3FZ8jY90KS1dBn/BxK7xOjmOQwcSNcQh/HazyE75P3Bbdk36esLq9IQS8jIGVLjkMIMfcjuFv4blIEVfx4/K1+V/x2I3mf5fZ7aSbdcGqHrjv0eq3WD9wy0XJ4oJVzeJziKu52FwxVsm6WPStCwb0bWSzgqPrHlq2MAGYNyZf5HxG+BieRpIfgeA+m7qNC2GOKdG4crU3nkcje4Yw46o65FmEuyoow5mV6kct4bl64xW2X+gR5n1AA1rFIU/mZ+Zeq1LoLTIBX8V1AefOri+LagVHKe2NwQRo3picAkNDma7tzXDe9aYodAc495aBuEWQH4eo5pBvH9zVcBnPqFS/C5TjHByzlxHdOwQYX8uVYq2Fx2w+A+qCBX3JpxUcRdruhrPYaqDxOc2u4g2bVhge5pgSu+ONsWREEItBdtnWOpyz8ZNlOH605sKuT5b1wb2S/ER4VvIwutOpX+CGvYWVWRCI33uoTBPx2Yl1HZ6JsKzLylJccV0Yoi8LOjavVnD0ETEaTLYcLlGA3Aiy0uQtCs4vzOdfMo7u6n7T/kXa6wziAvFslQ79ZFkf3CugJsKzkofRnRL9Wrx2lWUiPkuwtqBNhKWVj7GIlmbggY2gdXW4dW5JWSbg1QqOkuI35zwLpJLYfCCfhVngeD51iGNKHOtMXfxwbX5ues6o4QG4R83fcWIb9dvA6xxF3oh1DkAW89ygWasObzi3atGs4Ohb6GYMM1yUtw5pBBsjisLxTTmfVNEtxzlDwSnMEcLLFmo8scK2D+4KcFxk8aB+iYZ84dNyOB7HFVvHJop4EOtE+XlOtrMuWNHXCo6+YrOFSSNHw7g0iIh7H/Zj1/s14uKNyzHN5bkIn8Gf4njVf+oMwA2YcmCgRD9q+SpnC5xznOwmj59wvwTrhNn7SHpAXbCirxUcxULHMcwDXPEdxH/Atplrhy0H3DmliHE/4KP7C4EYz/mXX3EOb+w5XB/cVfCBCz5AOMQRx3V/Io4PpGZ6ThUQmzPp1A942VugxjSQcQ4mU/2EuJq9iE6sm4s63lGU27quxXUBZTGhrxUc8S4p0Xj2RdAAcpuXPrGso22t4RmtYDMlZIlPS1hmkmOSbK3wWgNHOoY5CZkFibLFUrPV0gXJGp4uvNaPW+LTEhbruvXBZ4XXyXH8DzVuwWJCsxK9AAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$\\displaystyle \\frac{a}{a b + a + 1} + \\frac{b}{b c + b + 1} + \\frac{c}{a c + c + 1}$"
      ],
      "text/plain": [
       "     a             b             c     \n",
       "─────────── + ─────────── + ───────────\n",
       "a⋅b + a + 1   b⋅c + b + 1   a⋅c + c + 1"
      ]
     },
     "execution_count": 94,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e4=e1+e2+e3;e4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 95,
   "id": "dafe65ae-127b-4d30-94d8-1728f3507311",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAgAAAAOCAYAAAASVl2WAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAXUlEQVQYGWP8//8/Aww0NDQYAdmrgdgYyP4AEmcBMgSA9GwgfgfEJkCsBMRwAFIAUhkKEgGyy4AUyBQ4YIKzcDBGFUAChmA4MIIiCxjEoAgCxQkoLkD0PSA+B8SnAeDgGGsGGU5UAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$\\displaystyle 1$"
      ],
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 95,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simplify(e4.subs(a,1/(b*c)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 96,
   "id": "b4dd0cd6-b998-4c82-a21c-a15a7243ea63",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{a}{a b + a + 1} + \\frac{b}{b c + b + 1} + \\frac{c}{a c + c + 1}\\;=\\;1\\;$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sE([e1+e2+e3,'=',1])"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.6"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}