Solving polynomials

Solving polynomials.ipynb

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  "cells": [
    {
      "metadata": {
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      "cell_type": "code",
      "source": "import sympy as sym\nfrom IPython.display import display, Math",
      "execution_count": 1,
      "outputs": []
    },
    {
      "metadata": {
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      "cell_type": "code",
      "source": "def disp(expr):\n    display(Math(sym.latex(expr)))",
      "execution_count": 22,
      "outputs": []
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "# Adding polynomials"
    },
    {
      "metadata": {
        "trusted": true
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      "cell_type": "code",
      "source": "x = sym.symbols('x')\n\np1 = 2*x**3 + x**2 - x\np2 = x**3 - x**4 - 4*x**2\ndisplay(Math('(%s) + (%s) = %s' %(sym.latex(p1), sym.latex(p2), sym.latex(p1+p2))))",
      "execution_count": 23,
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          "output_type": "display_data",
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            "text/plain": "<IPython.core.display.Math object>",
            "text/latex": "$\\displaystyle (2 x^{3} + x^{2} - x) + (- x^{4} + x^{3} - 4 x^{2}) = - x^{4} + 3 x^{3} - 3 x^{2} - x$"
          },
          "metadata": {}
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      ]
    },
    {
      "metadata": {
        "trusted": true
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      "cell_type": "code",
      "source": "p1 = sym.Poly(2*x**3 + x**2 - x)\ndisplay(p1)",
      "execution_count": 24,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": "Poly(2*x**3 + x**2 - x, x, domain='ZZ')",
            "text/latex": "$\\displaystyle \\operatorname{Poly}{\\left( 2 x^{3} + x^{2} - x, x, domain=\\mathbb{Z} \\right)}$"
          },
          "metadata": {}
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      ]
    },
    {
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      "cell_type": "code",
      "source": "p1.eval(2)",
      "execution_count": 25,
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 25,
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            "text/plain": "18",
            "text/latex": "$\\displaystyle 18$"
          },
          "metadata": {}
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    {
      "metadata": {
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      "cell_type": "code",
      "source": "p1.coeffs()",
      "execution_count": 26,
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 26,
          "data": {
            "text/plain": "[2, 1, -1]"
          },
          "metadata": {}
        }
      ]
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "# Multiplying polynomials"
    },
    {
      "metadata": {
        "trusted": true
      },
      "cell_type": "code",
      "source": "p1 = 4*x**2 - 2*x\np2 = x**3 - 1\ndisp(sym.expand(p1*p2))\ndisp(p1*p2)",
      "execution_count": 37,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": "<IPython.core.display.Math object>",
            "text/latex": "$\\displaystyle 4 x^{5} - 2 x^{4} - 4 x^{2} + 2 x$"
          },
          "metadata": {}
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": "<IPython.core.display.Math object>",
            "text/latex": "$\\displaystyle \\left(4 x^{2} - 2 x\\right) \\left(x^{3} - 1\\right)$"
          },
          "metadata": {}
        }
      ]
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "# Divide polynomials"
    },
    {
      "metadata": {
        "trusted": true
      },
      "cell_type": "code",
      "source": "p1 = 4*x**5 - x\np2 = 2*x**3 - x\ndisplay(Math('\\\\frac{%s}{%s} = %s' %(sym.latex(p1), sym.latex(p2), sym.latex(sym.simplify(p1/p2)))))",
      "execution_count": 39,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": "<IPython.core.display.Math object>",
            "text/latex": "$\\displaystyle \\frac{4 x^{5} - x}{2 x^{3} - x} = 2 x^{2} + 1$"
          },
          "metadata": {}
        }
      ]
    },
    {
      "metadata": {
        "trusted": true
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      "cell_type": "code",
      "source": "x,y = sym.symbols('x y')\nnums = range(5,16)\np1 = x**6 + (2*x**4) + (6*x) -y\np2 = x**3 + 3\nfor num in nums:\n      solution = sym.simplify((p1/p2).subs(y,num))\n      hasDenominator = sym.fraction(solution)[1] == 1\n      display(Math('\\\\text{Input: }%g\\\\text{, Solution:}%s\\\\text{ , has denominator: %s}' %(num,sym.latex(solution),hasDenominator)))",
      "execution_count": 64,
      "outputs": [
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            "text/plain": "<IPython.core.display.Math object>",
            "text/latex": "$\\displaystyle \\text{Input: }5\\text{, Solution:}\\frac{x^{6} + 2 x^{4} + 6 x - 5}{x^{3} + 3}\\text{ , has denominator: False}$"
          },
          "metadata": {}
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": "<IPython.core.display.Math object>",
            "text/latex": "$\\displaystyle \\text{Input: }6\\text{, Solution:}\\frac{x^{6} + 2 x^{4} + 6 x - 6}{x^{3} + 3}\\text{ , has denominator: False}$"
          },
          "metadata": {}
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": "<IPython.core.display.Math object>",
            "text/latex": "$\\displaystyle \\text{Input: }7\\text{, Solution:}\\frac{x^{6} + 2 x^{4} + 6 x - 7}{x^{3} + 3}\\text{ , has denominator: False}$"
          },
          "metadata": {}
        },
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          "output_type": "display_data",
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            "text/plain": "<IPython.core.display.Math object>",
            "text/latex": "$\\displaystyle \\text{Input: }8\\text{, Solution:}\\frac{x^{6} + 2 x^{4} + 6 x - 8}{x^{3} + 3}\\text{ , has denominator: False}$"
          },
          "metadata": {}
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": "<IPython.core.display.Math object>",
            "text/latex": "$\\displaystyle \\text{Input: }9\\text{, Solution:}x^{3} + 2 x - 3\\text{ , has denominator: True}$"
          },
          "metadata": {}
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            "text/plain": "<IPython.core.display.Math object>",
            "text/latex": "$\\displaystyle \\text{Input: }10\\text{, Solution:}\\frac{x^{6} + 2 x^{4} + 6 x - 10}{x^{3} + 3}\\text{ , has denominator: False}$"
          },
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            "text/plain": "<IPython.core.display.Math object>",
            "text/latex": "$\\displaystyle \\text{Input: }11\\text{, Solution:}\\frac{x^{6} + 2 x^{4} + 6 x - 11}{x^{3} + 3}\\text{ , has denominator: False}$"
          },
          "metadata": {}
        },
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          "output_type": "display_data",
          "data": {
            "text/plain": "<IPython.core.display.Math object>",
            "text/latex": "$\\displaystyle \\text{Input: }12\\text{, Solution:}\\frac{x^{6} + 2 x^{4} + 6 x - 12}{x^{3} + 3}\\text{ , has denominator: False}$"
          },
          "metadata": {}
        },
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          "output_type": "display_data",
          "data": {
            "text/plain": "<IPython.core.display.Math object>",
            "text/latex": "$\\displaystyle \\text{Input: }13\\text{, Solution:}\\frac{x^{6} + 2 x^{4} + 6 x - 13}{x^{3} + 3}\\text{ , has denominator: False}$"
          },
          "metadata": {}
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": "<IPython.core.display.Math object>",
            "text/latex": "$\\displaystyle \\text{Input: }14\\text{, Solution:}\\frac{x^{6} + 2 x^{4} + 6 x - 14}{x^{3} + 3}\\text{ , has denominator: False}$"
          },
          "metadata": {}
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": "<IPython.core.display.Math object>",
            "text/latex": "$\\displaystyle \\text{Input: }15\\text{, Solution:}\\frac{x^{6} + 2 x^{4} + 6 x - 15}{x^{3} + 3}\\text{ , has denominator: False}$"
          },
          "metadata": {}
        }
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