06 Polynomial Functions.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "fb1e0c82-5172-459b-847d-a62f8b5c99a8",
   "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": "57b9de1f-e246-484b-ac01-19c9f49f573f",
   "metadata": {},
   "source": [
    " <img src=\"images/linep.png\" alt=\"ml\" style=\"width : 100%;\"/>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cffc6021-bd72-4a64-9dcb-ff1d23f4d66e",
   "metadata": {},
   "source": [
    " <img src=\"images/p36.png\" alt=\"ml\" style=\"width : 100%;\"/>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "dd4387bc-1933-4ab0-88cd-c645cd153d12",
   "metadata": {},
   "source": [
    " <img src=\"images/lines.png\" alt=\"ml\" style=\"width : 100%;\"/>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "id": "e53b291d-7e36-4391-ae4a-0a4a5daf07c3",
   "metadata": {},
   "outputs": [],
   "source": [
    "def fp(n):\n",
    "    return(kpow(x,n))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "id": "4351f012-3b9a-483f-b806-a4d09ede9cae",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle We\\;will\\;take\\;P(-1)=P(\\;x^{3} - x^{2}\\;)\\;then\\;-1=\\;x^{3} - x^{2}\\;$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sE(['We  will take P(-1)=P(',x**3-x**2,') then -1=',x**3-x**2])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "id": "bac2f864-d184-483f-a67f-42ed1012945f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle x^{3} - x^{2}=-1$"
      ],
      "text/plain": [
       "x**3 - x**2=-1"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P=polyclass(x**3-x**2,-1);P.Q"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "id": "4391479a-096d-4933-8bc5-4bfe5a4a4446",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAADgAAAATCAYAAADF7c7rAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAB0UlEQVRYCdWW7VHCQBCGA0MBaAnYgR8dYAcwVgB2IMMv+IsdOJagJWgFIB1oCQwd4PPESyYyzAgHhrAzy242l82+7Oa9q61Wq2RXGY/HTZ4ZhudawfaIL3fN9d/rG5EvmADmPnsW/wn/A73IYlWx9chC+oBqF56d4LeIXRZilXBjAdq9WSUQ/FFELeYbXM9J5+xgB1u5EY39BnOMYSw7BK7y4JGcUMuLteCnhLcXQJLIoHYvT1g2NmqQ0Z/RBXqNZqyOmyTRADNw2FsTYSWYJfbL67KE99mpru/Df8D8IroogCTyX3JrGOBnCSWeAVopiQIIAvc8R0ObC2DzvTEPHtnJAVKcBfdDPTfYHmqn7kJsyppXfexZiFXe1AsVejp5VIlNUT/cNteOneAlk5OTFCAg7Nw6AKk/7Rj2fMN9QtWXbERngCyyn8w4z2LYlKX2hUMeJ+Ed1W4rXZ6bb7t4fV0KcEOCNgsPzoi8Z0neUg8ExW8wBU8RglOy8UyINdWf8Gn9NkLhHm8kmTdsSvX4xZEdcn3wjpbxV9lBO6YuAliPPI5SKsS8J6uepNRGo5GjJ4N+igBAbhUyqETjRi7wfFy5rpxQnxMoDs+iWqdPYpp+A1XYj0QQtzElAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$\\displaystyle x^{2} - 1$"
      ],
      "text/plain": [
       " 2    \n",
       "x  - 1"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e2=x*x-1;e2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "id": "4af944f0-b364-4794-bf50-34d7b2ae98ff",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle x^{7} + x^{6} + x^{5} + x^{3} + 2 x^{2} + 2 x$"
      ],
      "text/plain": [
       " 7    6    5    3      2      \n",
       "x  + x  + x  + x  + 2⋅x  + 2⋅x"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M=fp(7)+fp(6)+fp(5)+fp(3)+2*fp(2)+2*x;M"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "id": "6dbec936-8ea9-425d-8cd1-69d21f1e3899",
   "metadata": {},
   "outputs": [],
   "source": [
    "# e7  or x**7= x**3* x**3 *x -> e2*e2*x\n",
    "# e6  or x**6= x**3* x**3    -> e2*e2 \n",
    "# ......\n",
    "# ...\n",
    "e7,e6,e5,e4,e3=e2*e2*x,e2*e2,e2*x*x,e2*x,e2 # asigne all eeeessss"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "id": "b5aa3827-f4dc-4f83-994b-7f9141284685",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle whit\\;e\\;7\\;M=\\;x^{6} + 2 x^{5} - x^{3} + 2 x^{2} + 3 x\\;$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle whit\\;e\\;6\\;M=\\;2 x^{5} + x^{4} - x^{3} + 3 x + 1\\;$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle whit\\;e\\;5\\;M=\\;3 x^{4} - x^{3} - 2 x^{2} + 3 x + 1\\;$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle whit\\;e\\;4\\;M=\\;2 x^{3} - 2 x^{2} + 1\\;$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle whit\\;e\\;3\\;M=\\;-1\\;$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "vec=[e7,e6,e5,e4,e3]\n",
    "kk=7\n",
    "for i in vec:\n",
    "    i2=expand(i)\n",
    "    M=expand(M.subs(x**kk,i2))\n",
    "    sE(['whit  e',str(kk),'M=',M])\n",
    "    kk-=1 "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6b953341-9b2d-4b61-9775-f0e1d9006a99",
   "metadata": {},
   "source": [
    " <img src=\"images/linep.png\" alt=\"ml\" style=\"width : 100%;\"/>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7c9f27b7-ccfe-411f-8e6e-6b4fc4c602b0",
   "metadata": {},
   "source": [
    " <img src=\"images/p38.png\" alt=\"ml\" style=\"width : 100%;\"/>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b139b0ac-dc75-4fad-b5d2-b27d2c77ef66",
   "metadata": {},
   "source": [
    " <img src=\"images/lines.png\" alt=\"ml\" style=\"width : 100%;\"/>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "id": "46486d04-e4dc-46a8-bdbc-48355698e0a4",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle if\\;f(y)=2f(y-1)\\;then\\;..\\;f(y+1)=2f(y)...\\;also\\;$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle f(x)=2f(x-1)\\;then\\;..\\;f(x+1)=2f(x)\\;$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle then\\;f(x+1)=2f(x)\\;and\\;..\\;f(x+2)=2f(x+1)\\;$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sE(['if','f(y)=2f(y-1)','then ..','f(y+1)=2f(y)...  also'])\n",
    "sE(['f(x)=2f(x-1)','then ..','f(x+1)=2f(x)'])\n",
    "sE(['then','f(x+1)=2f(x)','and ..','f(x+2)=2f(x+1)'])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "id": "060fd37f-0165-48d0-b25a-dcc860235eb2",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle f(x+2)=x\\;+f(x)+f(x+1)\\;data\\;from\\;problem\\;$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle replace\\;f(x+2)\\;by\\;2f(x+1)\\;$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sE(['f(x+2)=x +f(x)+f(x+1)  data from problem']) \n",
    "sE(['replace  f(x+2) by 2f(x+1)']) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "id": "cc0c4748-8175-43ad-97c3-63528092481d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 2f(x+1)=x\\;+f(x)+f(x+1)\\;$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sE(['2f(x+1)=x +f(x)+f(x+1)'])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "id": "c5151ca1-f72a-428e-8d9b-16f252cc538b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle f(x+1)=x\\;+f(x)\\;$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sE(['f(x+1)=x +f(x)'])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "id": "4a192423-0b83-4d8f-a82a-84450bab891d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle replace\\;f(x+1)=2f(x)\\;$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 2f(x)=x\\;+f(x)\\;$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sE(['replace  f(x+1)=2f(x)'])\n",
    "sE(['2f(x)=x +f(x)'])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "id": "5082b1df-e584-42f6-8ed8-4aeb0d154808",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle find\\;that\\;f(x)=x\\;$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sE([' find that f(x)=x'])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "id": "cd42c299-0438-4b23-825b-d463ce14baaa",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle f(-3)+f(4)=\\;-3+4\\;=\\;1\\;$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sE(['f(-3)+f(4)= -3+4 =',1])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ef255d21-01ae-4515-b20e-3a69e78e923e",
   "metadata": {},
   "source": [
    " <img src=\"images/linep.png\" alt=\"ml\" style=\"width : 100%;\"/>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "472e9610-2269-4ff8-afc0-e377cb7ca606",
   "metadata": {},
   "source": [
    " <img src=\"images/p39.png\" alt=\"ml\" style=\"width : 100%;\"/>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b5e02550-1461-43b0-a8e0-b9845a4b8a2b",
   "metadata": {},
   "source": [
    " <img src=\"images/lines.png\" alt=\"ml\" style=\"width : 100%;\"/>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "5e4c8845-6569-4619-9f07-8b847eece767",
   "metadata": {},
   "outputs": [],
   "source": [
    "def fx(x):\n",
    "    return(exp(x)+kpow(pi,x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "6a12245a-eea9-4d89-bdd2-b5888f675584",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle e + \\pi$"
      ],
      "text/plain": [
       "ℯ + π"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fx(1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "f6ca344d-789d-41eb-9dd1-821d3c935d23",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle e^{3} + \\pi^{3}$"
      ],
      "text/plain": [
       " 3    3\n",
       "ℯ  + π "
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fx(3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "5a047363-238b-48d4-a048-d1bbb470f564",
   "metadata": {},
   "outputs": [],
   "source": [
    "eq1=1-kpow(pi,3);eq2=1-exp(3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "12a88de0-5157-47db-aa13-9cdbeee2ceda",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle we\\;have\\;the\\;followin\\;data:\\;$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle e^{3}\\;=\\;1 - \\pi^{3}\\;and\\;\\pi^{3}\\;=\\;1 - e^{3}\\;$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sE([' we have the followin data:'])\n",
    "sE([exp(3),'=',eq1,' and ',kpow(pi,3),'=',eq2])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "279d9e76-2677-47cd-80bd-7703d4d289b7",
   "metadata": {},
   "source": [
    " <img src=\"images/p39.png\" alt=\"ml\" style=\"width : 100%;\"/>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "a89f6573-ee9c-4421-bfd3-33984478d005",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - \\pi^{7} - e^{7} + e^{4} + \\pi^{4}$"
      ],
      "text/plain": [
       "   7    7    4    4\n",
       "- π  - ℯ  + ℯ  + π "
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M=fx(4)-fx(7);M"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "8f56a8e8-fabe-4dd1-9ff1-e878f2b64c8e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[ \\pi^{4}, \\  - \\pi^{7}, \\  - e^{7}, \\  e^{4}\\right]$"
      ],
      "text/plain": [
       "⎡ 4    7    7   4⎤\n",
       "⎣π , -π , -ℯ , ℯ ⎦"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fpoly(M,'list')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "75ff3dfc-f795-4e8f-95d2-5e6576359b91",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\pi^{4} \\left(1 - \\pi^{3}\\right)$"
      ],
      "text/plain": [
       " 4 ⎛     3⎞\n",
       "π ⋅⎝1 - π ⎠"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e1=pi**4*(opemat(fpoly(M,'gets','01')/pi**4,'s'));e1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "283bbc19-76a8-4cee-bf90-a1e9cc82c784",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left(1 - e^{3}\\right) e^{4}$"
      ],
      "text/plain": [
       "⎛     3⎞  4\n",
       "⎝1 - ℯ ⎠⋅ℯ "
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e2=exp(1)**4*(opemat(fpoly(M,'gets','23')/exp(1)**4,'s'));e2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "8a77519b-62fa-45b0-9663-aafe1a57cb02",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\pi^{4} \\left(1 - \\pi^{3}\\right) + \\left(1 - e^{3}\\right) e^{4}$"
      ],
      "text/plain": [
       " 4 ⎛     3⎞   ⎛     3⎞  4\n",
       "π ⋅⎝1 - π ⎠ + ⎝1 - ℯ ⎠⋅ℯ "
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e3=e1+e2;e3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "40320552-1051-4a55-9bd0-040fa27b8bc9",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left(1 - e^{3}\\right) e^{4} + \\pi^{4} e^{3}$"
      ],
      "text/plain": [
       "⎛     3⎞  4    4  3\n",
       "⎝1 - ℯ ⎠⋅ℯ  + π ⋅ℯ "
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e3=e3.subs(eq1,exp(3));e3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "4dae7a4d-1f28-4bb8-b648-7d43d2ed3ff9",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\pi^{3} e^{4} + \\pi^{4} e^{3}$"
      ],
      "text/plain": [
       " 3  4    4  3\n",
       "π ⋅ℯ  + π ⋅ℯ "
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e3=e3.subs(eq2,pi**3);e3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "a6d94181-d28f-4267-b2c8-3fd2bb7ca054",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\pi^{3} \\left(e + \\pi\\right) e^{3}$"
      ],
      "text/plain": [
       " 3          3\n",
       "π ⋅(ℯ + π)⋅ℯ "
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e3=factor(e3);e3"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "de7543e2-4a0b-468e-a9ae-e59504bcd4b5",
   "metadata": {},
   "source": [
    " <img src=\"images/p39.png\" alt=\"ml\" style=\"width : 100%;\"/>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "aaa72527-b813-40e9-9e93-20111487f65c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACkAAAArCAYAAADlqKH9AAAACXBIWXMAAA7EAAAOxAGVKw4bAAACeElEQVRYCe2Y0XETMRRFbcYFmKGD0EECFQAdhKECSAdk8mX/eUgHCRWEpIOEChhSQqiAjDsw5ziSZ3cT21pZfDDozVxLq326e/2epNVquFgsBiVtOp3uw3cJDqjPS3CPSpAgZgzPV3APXoE9UMxKiTRi71WF4M8URrOYPSvG9BeJqshSwa2RrJEsFYFSPHVM/leRHJbaYPA6dFPhO9x3t+UduAU/uHdKmW3FRGYrSOhYJ05CkJJcaiSTwpTgVCOZEKQklxrJpDAlOA0nk0nZb9qEh/Z1qW+cvhFb5/9PTJzswwF2Nnv880PgwYC7ntdgRrs7n162jStbJCq+gLu4DaP05OI7eA762kauXUTOOkpecO0eMsc2cmWLfCKtpt6I9LZtXNkio5KQ5g9cn1E/j+055TquIusk5E4cPx+uqe/0qfAUVxGRRg1yj/t+Ag9Pe89wOaJ1uVbp5oaz8wQYlXV2jt8R0OcXeEM9CpqHTn6IxbZlU/CX+3fweUl5TPs8hWu5mOPogHd2ehAqTNkBOALHoW6EvB5QKugeNGfzW65t/wZWhq/tRviC+qmgfgYUPeB6K9cIJ9PkZ+dV6GREnQSugT7ghrIVGf0w/8wJ92J0XMxb5+Tcc8G/Bka/ySGv7dE2cj0ak5A5+N/Zm9LJ8JEypjKSJpX0U4iCzIZmtjSfcfNQ3f47arrQ0ag2Be3nCgy8CryCY6cZ391gOOYuwgMsTNeu1hy3WVwrkfzbMQyfQCsNtBuNXFsrEF7fUEnWTLcCb+ncTLd1x2dLeBLzg5OrxnJFiH1CMGwXSdYUqZhZp1euuCUNglxXx0BBcRUYcN0S3nnmo8s/NNH0RyD+PooAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$\\displaystyle \\frac{1}{\\pi^{3} e^{3}}$"
      ],
      "text/plain": [
       " -3\n",
       "ℯ  \n",
       "───\n",
       "  3\n",
       " π "
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e4=fx(1)/e3;e4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "4b9f8230-347f-4bd1-a03b-61ab4e736221",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{1}{e \\pi}$"
      ],
      "text/plain": [
       " -1\n",
       "ℯ  \n",
       "───\n",
       " π "
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e5=rpow(e4,3);e5"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "37010a5d-6dfb-4a3b-8050-4b8d50a423ac",
   "metadata": {},
   "source": [
    " <img src=\"images/p39.png\" alt=\"ml\" style=\"width : 100%;\"/>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "id": "c7bda7fc-9ab9-47fb-8639-41ea5c1cee4f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\sqrt[3]{e + \\pi} \\sqrt[3]{\\frac{1}{- \\pi^{7} - e^{7} + e^{4} + \\pi^{4}}}\\;=\\;\\frac{1}{e \\pi}\\;$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sE([rpow(((fx(1)/(fx(4)-fx(7)))),3),'=',e5])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "977880fb-268a-433c-a863-cbbefc9c07ac",
   "metadata": {},
   "source": [
    " <img src=\"images/linep.png\" alt=\"ml\" style=\"width : 100%;\"/>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f2168a7c-c748-49c3-a8bc-291321e8fd6d",
   "metadata": {},
   "source": [
    " <img src=\"images/p40.png\" alt=\"ml\" style=\"width : 100%;\"/>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9854ce07-b259-4ce9-b9ac-54c377dbc604",
   "metadata": {},
   "source": [
    " <img src=\"images/lines.png\" alt=\"ml\" style=\"width : 100%;\"/>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "f505f445-7654-49f6-bd57-465da157abba",
   "metadata": {},
   "outputs": [],
   "source": [
    "x,n=symbols('x n')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "id": "46b6f092-d4c1-4bc3-9a26-1e03d6cb31fd",
   "metadata": {},
   "outputs": [],
   "source": [
    "def f(x):\n",
    "        kres=kpow(x+1,n)+kpow(x+2,n)+kpow(x+3,n)\n",
    "        kres=kpow(x+n-1,n)+kpow(x+n,n)+kpow(x+n+1,n)+kres\n",
    "        kres=kpow(x+2*n-3,n)+kpow(x+2*n-2,n)+kpow(x+2*n-1,n)+kres\n",
    "        return(kres)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "id": "f2c564f2-2338-49d7-94ba-59846df383c0",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left(-1\\right)^{n} + 0^{n} + \\left(1 - n\\right)^{n} + \\left(2 - n\\right)^{n} + \\left(3 - n\\right)^{n} + \\left(n - 3\\right)^{n} + \\left(n - 2\\right)^{n} + \\left(n - 1\\right)^{n} + 1$"
      ],
      "text/plain": [
       "    n    n          n          n          n          n          n          n  \n",
       "(-1)  + 0  + (1 - n)  + (2 - n)  + (3 - n)  + (n - 3)  + (n - 2)  + (n - 1)  +\n",
       "\n",
       "  \n",
       " 1"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "kk=f(-n);kk"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "id": "458bd404-5c8a-48d9-8d96-e6149ddce334",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOCAYAAAAWo42rAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA0UlEQVQoFXWR0Q2CQBBEhVgAwQ60BLUCsQO0A+3BP/4MJWgLWgIdGCmBDiB2gG+OW0PIuckyO3Ozu5cj6vt+ZlEURenrFlyRJVojLTIjwht+BZ86ABNA2p66ib14BhMzee0Dqukm7ozggawlTOIFzzTdjBlCNzGJaqoii/1dBvr/m2pi6s+te2y3Lb/V48NQvdBE6woZbFurO9rKJOA0bXhHDBW5DBhtYmXP88C0CRjXaLW2OiPFHaEDczNTa+2RPEmb6+ND3RcMdoUtfAd3f+wLrOY/XeyA6s8AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$\\displaystyle 0$"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "kk.subs(n,1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "id": "324e0b8d-ab3b-4f84-9ddf-2b482600e3c2",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOCAYAAAAWo42rAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA0UlEQVQoFXWR0Q2CQBBEhVgAwQ60BLUCsQO0A+3BP/4MJWgLWgIdGCmBDiB2gG+OW0PIuckyO3Ozu5cj6vt+ZlEURenrFlyRJVojLTIjwht+BZ86ABNA2p66ib14BhMzee0Dqukm7ozggawlTOIFzzTdjBlCNzGJaqoii/1dBvr/m2pi6s+te2y3Lb/V48NQvdBE6woZbFurO9rKJOA0bXhHDBW5DBhtYmXP88C0CRjXaLW2OiPFHaEDczNTa+2RPEmb6+ND3RcMdoUtfAd3f+wLrOY/XeyA6s8AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$\\displaystyle 0$"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "kk.subs(n,3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "id": "2fdb77e0-5b5a-4d86-ba2a-ec75034f42e1",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOCAYAAAAWo42rAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA0UlEQVQoFXWR0Q2CQBBEhVgAwQ60BLUCsQO0A+3BP/4MJWgLWgIdGCmBDiB2gG+OW0PIuckyO3Ozu5cj6vt+ZlEURenrFlyRJVojLTIjwht+BZ86ABNA2p66ib14BhMzee0Dqukm7ozggawlTOIFzzTdjBlCNzGJaqoii/1dBvr/m2pi6s+te2y3Lb/V48NQvdBE6woZbFurO9rKJOA0bXhHDBW5DBhtYmXP88C0CRjXaLW2OiPFHaEDczNTa+2RPEmb6+ND3RcMdoUtfAd3f+wLrOY/XeyA6s8AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$\\displaystyle 0$"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "kk.subs(n,81)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d38b457a-6ce5-4c90-8dd3-c13f68f58040",
   "metadata": {},
   "source": [
    " <img src=\"images/linep.png\" alt=\"ml\" style=\"width : 100%;\"/>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f9fa6896-2e2c-40b9-b32c-a092bf7a3d74",
   "metadata": {},
   "source": [
    " <img src=\"images/p45.png\" alt=\"ml\" style=\"width : 100%;\"/>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e297d6bf-07e7-4aad-a2f6-58d8c0908160",
   "metadata": {},
   "source": [
    " <img src=\"images/lines.png\" alt=\"ml\" style=\"width : 100%;\"/>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "09401fc8-0dd8-461a-b89d-992a8dd754d8",
   "metadata": {},
   "outputs": [],
   "source": [
    "def fx(x):\n",
    "    return((x+1)/(x-1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "43dbb900-75b4-47ce-8b1e-4d680dd70577",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle From\\;data\\;f(1/2)=\\;-3\\;.f(\\;-3\\;)+3\\;$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sE(['From','data','f(1/2)=',fx(frs(1,2)),'.f(',fx(frs(1,2)),')+3'])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "398227a5-81d9-4a95-957a-478c4617be64",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle Also..\\;f(-3)=\\;0.5\\;.f(\\;0.5\\;)+3\\;$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sE(['Also..','f(-3)=',fx(-3),'.f(',fx(-3),')+3'])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "7c3590c3-09aa-4434-abc0-d586216a1f88",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{x}{2} + 3$"
      ],
      "text/plain": [
       "x    \n",
       "─ + 3\n",
       "2    "
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# f(-3)  = y\n",
    "# f(1/2) = x then\n",
    "x,y = symbols('x y')\n",
    "csolve(y-x/2-3,y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "afd26738-6d3c-44a7-a386-7eb72b4f545a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - \\frac{12}{5}$"
      ],
      "text/plain": [
       "-12/5"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "csolve(x -(-3*(x/2+3)+3),x)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e092ee94-3762-4f1f-a23c-f8841c23fb70",
   "metadata": {},
   "source": [
    " <img src=\"images/linep.png\" alt=\"ml\" style=\"width : 100%;\"/>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "916c1537-102e-4f35-bcb4-8a9dcabefb6a",
   "metadata": {},
   "source": [
    " <img src=\"images/p47.png\" alt=\"ml\" style=\"width : 100%;\"/>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "20246984-bc28-4f0d-a654-4bd522f52caf",
   "metadata": {},
   "source": [
    " <img src=\"images/lines.png\" alt=\"ml\" style=\"width : 100%;\"/>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "08e18b62-6af8-4af4-b3d3-4e2b66f50e0e",
   "metadata": {},
   "outputs": [],
   "source": [
    "a,b,c,x=symbols('a b c x')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "8819a0fc-fa09-4297-8af0-7d69d76f09c6",
   "metadata": {},
   "outputs": [],
   "source": [
    "def P(x):\n",
    "    return(opemat(x*x+a,'esfs'))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "0251885b-2972-4c93-a715-3c93c2bcc202",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle a^{4} + 4 a^{3} x^{2} + 2 a^{3} + 6 a^{2} x^{4} + 4 a^{2} x^{2} + a^{2} + 4 a x^{6} + 2 a x^{4} + a + x^{8}$"
      ],
      "text/plain": [
       " 4      3  2      3      2  4      2  2    2        6        4        8\n",
       "a  + 4⋅a ⋅x  + 2⋅a  + 6⋅a ⋅x  + 4⋅a ⋅x  + a  + 4⋅a⋅x  + 2⋅a⋅x  + a + x "
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a1=P(P(P(x)));a1 # n=3"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c2580920-e176-448e-926b-ad9ae729a9b7",
   "metadata": {},
   "source": [
    " <img src=\"images/p47a.png\" alt=\"ml\" style=\"width : 80%;\"/>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "4b8e8db2-2bcf-44c8-b89e-ea9a8831bdd1",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle a^{4} + 2 a^{3} + a^{2} + a$"
      ],
      "text/plain": [
       " 4      3    2    \n",
       "a  + 2⋅a  + a  + a"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fpoly(a1,'unfilt',x) # return al monom if dont have x..."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "4bd2d171-e4f7-4eeb-bda8-f68be9028013",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle if\\;a^{4} + 2 a^{3} + a^{2} + a\\;=\\;5\\;then\\;a\\;=\\;1\\;$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "show_exp(['if',fpoly(a1,'unfilt',x),'=',5,'then',a,'=',1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "8874aef5-c00f-4cb1-b950-07ac0c194ffc",
   "metadata": {},
   "outputs": [],
   "source": [
    "def P(x):\n",
    "    return(opemat(x*x+1,'esfs'))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "40d54979-5d6f-40b9-b095-8674c509e904",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle x^{8} + 4 x^{6} + 8 x^{4} + 8 x^{2} + 5$"
      ],
      "text/plain": [
       " 8      6      4      2    \n",
       "x  + 4⋅x  + 8⋅x  + 8⋅x  + 5"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P(P(P(x)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "1dfdac16-5102-4bbe-a24e-dfbdeecd5072",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 1226$"
      ],
      "text/plain": [
       "1226"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P(35)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "08a14fcd-8233-4e84-a1ef-18199ebc1d7e",
   "metadata": {},
   "source": [
    " <img src=\"images/linep.png\" alt=\"ml\" style=\"width : 100%;\"/>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cc22dd6a-d1b9-465d-8e7c-2cd2b9504be4",
   "metadata": {},
   "source": [
    " <img src=\"images/p48.png\" alt=\"ml\" style=\"width : 100%;\"/>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "69b2d40e-60de-4384-b62f-887acc214879",
   "metadata": {},
   "source": [
    " <img src=\"images/lines.png\" alt=\"ml\" style=\"width : 100%;\"/>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "aab24d18-7a58-4418-9147-13271305b42c",
   "metadata": {},
   "source": [
    " <img src=\"images/p48a.png\" alt=\"ml\" style=\"width : 50%;\"/>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "id": "f15992d9-c7ca-44e8-92f3-e13349e51db7",
   "metadata": {},
   "outputs": [],
   "source": [
    "a=rpow(3*x);b=rpow(x+2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "id": "af6990c9-9607-4830-8af4-5863f61a1bc1",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 2 \\sqrt{3} \\sqrt{x} \\sqrt{x + 2} + 4 x + 2$"
      ],
      "text/plain": [
       "          _______          \n",
       "2⋅√3⋅√x⋅╲╱ x + 2  + 4⋅x + 2"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e1=expand(kpow(a+b,2));e1 #pow a+b to 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "3e9f4ebb-c452-4053-8486-fe88cd4636cb",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\sqrt{3} \\sqrt{x} \\sqrt{x + 2} + 2 x + 1$"
      ],
      "text/plain": [
       "        _______          \n",
       "√3⋅√x⋅╲╱ x + 2  + 2⋅x + 1"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e2=e1/2;e2 #divide by 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "7f7fec41-b9b5-4d21-acda-f27b40c61cd4",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\sqrt{3} \\sqrt{x} \\sqrt{x + 2} + 2 x$"
      ],
      "text/plain": [
       "        _______      \n",
       "√3⋅√x⋅╲╱ x + 2  + 2⋅x"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e3=e2-1;e3 #Add -1 and get tha same eq from question "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "id": "69671396-0fc4-4164-b129-b5332b75ffdd",
   "metadata": {},
   "outputs": [],
   "source": [
    "# creatinf func whit the same step\n",
    "def f(y):\n",
    "    return(kpow(y,2)/2-1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "id": "8dec6e5b-a45e-46bf-89c3-58dac633c905",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{x^{2}}{2} - 1$"
      ],
      "text/plain": [
       " 2    \n",
       "x     \n",
       "── - 1\n",
       "2     "
      ]
     },
     "execution_count": 80,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(x) # whi x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "id": "7357d837-62f5-4446-a12b-6f60d1326f95",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\sqrt{3} \\sqrt{x} + \\sqrt{x + 2}$"
      ],
      "text/plain": [
       "          _______\n",
       "√3⋅√x + ╲╱ x + 2 "
      ]
     },
     "execution_count": 81,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a+b # a+b ?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "id": "58b6074d-8094-4d3a-bf4f-992d9ad7c59b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\sqrt{3} \\sqrt{x} \\sqrt{x + 2} + 2 x$"
      ],
      "text/plain": [
       "        _______      \n",
       "√3⋅√x⋅╲╱ x + 2  + 2⋅x"
      ]
     },
     "execution_count": 82,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simplify(expand(f(a+b))) # test the func ok"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b4c3cb04-6ef6-402b-9abe-002311fe6d94",
   "metadata": {},
   "source": [
    " <img src=\"images/p48b.png\" alt=\"ml\" style=\"width : 50%;\"/>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "id": "9b088f73-790f-4ca9-81e5-2eef55a5a227",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{x^{2} + 4 x - 2}{2}$"
      ],
      "text/plain": [
       " 2          \n",
       "x  + 4⋅x - 2\n",
       "────────────\n",
       "     2      "
      ]
     },
     "execution_count": 83,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eq3=factor(expand(f(x)-f(x-1)+f(x+1)));eq3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "id": "4b382abf-a88c-4d46-8f32-e86bfb6d33fe",
   "metadata": {},
   "outputs": [],
   "source": [
    "g,G=symbols('g G')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9c5bc181-a42f-43bd-8c44-c0914f78a355",
   "metadata": {},
   "source": [
    " <img src=\"images/p48b.png\" alt=\"ml\" style=\"width : 50%;\"/>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "id": "b9d79d44-12ea-44fd-b444-39d7c57c1be0",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{x^{2} + 4 x - 2}{2 \\left(x + 2\\right)^{2}}=\\frac{G - 6}{2 G}$"
      ],
      "text/plain": [
       "(x**2 + 4*x - 2)/(2*(x + 2)**2)=(G - 6)/(2*G)"
      ]
     },
     "execution_count": 86,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P=polyclass(eq3/kpow(x+2,2),(G-6)/(2*G));P.Q"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 94,
   "id": "aac46645-f754-4c5c-8f71-9ddaff3c780c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 2 G x^{2} + 8 G x - 4 G$"
      ],
      "text/plain": [
       "     2              \n",
       "2⋅G⋅x  + 8⋅G⋅x - 4⋅G"
      ]
     },
     "execution_count": 94,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eq4=expand(numer(P.q1)*denom(P.q2));eq4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 95,
   "id": "17dd48d3-875f-4b88-aab8-24f0a2f3b327",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 2 G x^{2} + 8 G x + 8 G - 12 x^{2} - 48 x - 48$"
      ],
      "text/plain": [
       "     2                     2            \n",
       "2⋅G⋅x  + 8⋅G⋅x + 8⋅G - 12⋅x  - 48⋅x - 48"
      ]
     },
     "execution_count": 95,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eq5=expand(numer(P.q2)*denom(P.q1));eq5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 99,
   "id": "f62a7b72-5f3d-42c1-a3f4-b63f4a15fffb",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left(x + 2\\right)^{2}$"
      ],
      "text/plain": [
       "       2\n",
       "(x + 2) "
      ]
     },
     "execution_count": 99,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "factor(csolve((eq4-eq5)/12,G))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 100,
   "id": "0bca8b59-bc03-4e51-b3ba-c07e1dd5cfa1",
   "metadata": {},
   "outputs": [],
   "source": [
    "def g(x):\n",
    "    return(kpow(x+2,2))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "627afd73-3896-4a13-a45c-0165b2f386fc",
   "metadata": {},
   "source": [
    " <img src=\"images/p48c.png\" alt=\"ml\" style=\"width : 50%;\"/>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 119,
   "id": "8a91180c-75a5-434b-9b36-3f5a53ad8a75",
   "metadata": {},
   "outputs": [],
   "source": [
    "def M(x):\n",
    "    return( 2*( simple_m(rpow(x)  ,'O') )-1) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 120,
   "id": "e3b6f729-58df-4039-801b-f5846f873b39",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 2 x + 3$"
      ],
      "text/plain": [
       "2⋅x + 3"
      ]
     },
     "execution_count": 120,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M(g(x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 121,
   "id": "b82ca90c-9b50-4d89-a94e-d69dcdc2f9d2",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle M(x)=\\;2 \\sqrt{x} - 1\\;$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sE(['M(x)=',M(x)])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a0c61f9f-af8a-458f-a4e7-3d7ec0e7c3a4",
   "metadata": {},
   "source": [
    " <img src=\"images/linep.png\" alt=\"ml\" style=\"width : 100%;\"/>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3fde1f19-c623-445c-936b-ca2c9badd90c",
   "metadata": {},
   "source": [
    " <img src=\"images/p49.png\" alt=\"ml\" style=\"width : 100%;\"/>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8111a399-1aba-4e19-98e3-945c6efaaf28",
   "metadata": {},
   "source": [
    " <img src=\"images/lines.png\" alt=\"ml\" style=\"width : 100%;\"/>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "af80b428-0b29-452f-ac3f-d820eb636bd9",
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "x,B,a,b,n=symbols('x B a b n')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "5c4c418e-470a-4aaf-a84f-5ed53752b26e",
   "metadata": {},
   "outputs": [],
   "source": [
    "def P(x):\n",
    "    return(simplify(factor(a*x+b)))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b08ca69a-a200-4851-8b47-263156da6fa9",
   "metadata": {},
   "source": [
    " <img src=\"images/p49.png\" alt=\"ml\" style=\"width : 100%;\"/>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "ede84a25-010a-4468-862b-52a3f0a5c9ea",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - \\frac{a^{- n} b \\left(a^{n} - 1\\right)}{a - 1}$"
      ],
      "text/plain": [
       "  -n   ⎛ n    ⎞ \n",
       "-a  ⋅b⋅⎝a  - 1⎠ \n",
       "────────────────\n",
       "     a - 1      "
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B1= (-b*(kpow(a,n)-1))/(kpow(a,n)*(a-1));B1"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bae39746-f8ec-4490-8e42-246b54de925e",
   "metadata": {},
   "source": [
    " <img src=\"images/p49.png\" alt=\"ml\" style=\"width : 100%;\"/>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "711e15d5-a322-4f0a-98c8-32ea0b8251de",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{a^{- n} b \\left(a - a^{n}\\right)}{a - 1}$"
      ],
      "text/plain": [
       " -n   ⎛     n⎞\n",
       "a  ⋅b⋅⎝a - a ⎠\n",
       "──────────────\n",
       "    a - 1     "
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P(B1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "f0282b53-6065-488d-93b7-c67d1aebce0a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{a^{- n} b \\left(a^{2} - a^{n}\\right)}{a - 1}$"
      ],
      "text/plain": [
       " -n   ⎛ 2    n⎞\n",
       "a  ⋅b⋅⎝a  - a ⎠\n",
       "───────────────\n",
       "     a - 1     "
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P(P(B1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "860bdcc3-838f-4848-b6ef-a19d17c92ec6",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{a^{- n} b \\left(a^{3} - a^{n}\\right)}{a - 1}$"
      ],
      "text/plain": [
       " -n   ⎛ 3    n⎞\n",
       "a  ⋅b⋅⎝a  - a ⎠\n",
       "───────────────\n",
       "     a - 1     "
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P(P(P(B1)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "e419f1aa-9f88-42c6-bc38-850fe9029a2d",
   "metadata": {},
   "outputs": [],
   "source": [
    "def Pn(x):\n",
    "    return(a**-n*b*(a**x-a**n)/(a-1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "32641529-9051-403c-a9c5-f6bd56fc1090",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{a^{- n} b \\left(- a^{n} + a^{x}\\right)}{a - 1}$"
      ],
      "text/plain": [
       " -n   ⎛   n    x⎞\n",
       "a  ⋅b⋅⎝- a  + a ⎠\n",
       "─────────────────\n",
       "      a - 1      "
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Pn(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "6fae425b-9a50-456a-ba0f-61a814003cc0",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle P(P(P....P(B)...)) =0$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "show_res(Pn(n),'P(P(P....P(B)...))')"
   ]
  }
 ],
 "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
}