Lesson Solving algebraic equations of high degree
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<H2>Solving algebraic equations of high degree</H2> Some algebraic equations of high degree can be solved by reduction to the quadratic equation. Below are examples of three forms of such equations. Note that the lessons explaining how to solve quadratic equations are - <A HREF=http://www.algebra.com/algebra/homework/quadratic/lessons/proof-of-quadratic-by-completing-the-square.lesson>PROOF of quadratic formula...</A>, - <A HREF=http://www.algebra.com/algebra/homework/quadratic/lessons/Introduction-Into-Quadratics.lesson>Introduction into quadratic equations</A> and - <A HREF=http://www.algebra.com/algebra/homework/quadratic/lessons/Solving-quadratic-equations-without-quadratic-formula.lesson>Solving quadratic equations without quadratic formula</A> in the module <B>Quadratic equations</B> of the section <B>Algebra-I</B>. <H3>Example 1</H3>Solve the equation {{{x^4+5x^3+6x^2=0}}}. <B>Solution</B> The polynomial {{{x^4+5x^3+6x^2=0}}} can be factored into the product of the polynomials {{{x^2}}} and {{{(x^2+5x+6)}}}: {{{x^4+5x^3+6x^2 = x^2*(x^2+5x+6)}}}. Therefore, the polynomial {{{x^4+5x^3+6x^2}}} is equal to zero if and only if one of two multipliers {{{x^2}}} or {{{x^4+5x^3+6x^2}}} is equal to zero. So, to solve the equation {{{x^4+5x^3+6x^2=0}}}, we need to solve two equations {{{x^2=0}}} and {{{x^4+5x^3+6x^2=0}}} separately. Solve the equation {{{x^2=0}}} first. It has two equal roots: {{{x[1]=x[2]=0}}}. Then solve the equation {{{x^2+5x+6=0}}}. It has the roots {{{x[3]=-2}}} and {{{x[4]=-3}}}. Thus, the roots of the original equation are {{{x[1]=x[2]=0}}}, {{{x[3]=-2}}}, {{{x[4]=-3}}}. <H3>Example 2</H3>Solve the equation {{{x^3=8}}}. <B>Solution</B> Let us rewrite this equation as {{{x^3-8=0}}}. One root is {{{x[1]=2}}}. Factor left side of {{{x^3-8}}} into the product of {{{x-2}}} and the polynomial of the degree 2 {{{x^2+2x+4}}}, so {{{x^3-8=(x-2)*(x^2+2x+4)}}}. Solve the equation {{{x^2+2x+4=0}}} using the quadratic formula. The two roots are complex numbers {{{x[2]=-1+sqrt(-3)}}} and {{{x[3]=-1-sqrt(-3)}}}. Thus, the roots of the original equation are {{{x[1]=2}}}, {{{x[2]=-1+sqrt(-3)}}} and {{{x[3]=-1-sqrt(-3)}}}. <H3>Example 3</H3>Solve the equation {{{x^4-13x^2+36=0}}}. Let us introduce the new variable {{{z=x^2}}}. Then the equation takes the form {{{z^2-13z+36=0}}}. This is a quadratic equation. It has two roots {{{z[1]=9}}}, {{{z[2]=4}}}. Now solve the equation {{{x^2=9}}}. The roots of this equation are {{{x[1]=3}}} and {{{x[1]=-3}}}. Then solve the equation {{{x^2=4}}}. The roots of this equation are {{{x[1]=2}}} and {{{x[1]=-2}}}. So, the roots of the original equation are {{{x[1]=3}}}, {{{x[1]=-3}}}, {{{x[3]=2}}} and {{{x[4]=-2}}}. Equations of this form are called <B>bi-quadratic equations</B>. <H3>Example 4</H3>Solve an equation {{{x^6-16x^3+64=0}}}. <B>Solution</B> Let us introduce the new variable {{{z=x^3}}}. Then the equation takes the form {{{z^2-16z+64=0}}}. This is a quadratic equation. It has two equal roots {{{z[1]=z[2]=8}}}. Now solve the equation {{{x^3=8}}}. It has three roots {{{x[1]=2}}}, {{{x[2]=-1+sqrt(-3)}}} and {{{x[3]=-1-sqrt(-3)}}} (see <B>Example 2</B>). They are the solutions of the original equation. Other three roots of the original equation are equal to these three (because {{{z[1]=z[2]}}}). <H3>Notes</H3>Method of the <B>Example 1</B> can be extended to solve equations of the form {{{x^n*(ax^2+bx+c)=0}}}. Method of the <B>Example 2</B> can be extended to solve equations of the form {{{(ax+b)*(cx^2+dx+e)=0}}}. Method of the <B>Example 3</B> can be extended to solve equations of the form {{{ax^(2n)+bx^n+c=0}}}. My other closely related lessons on solving systems of non-linear equations in this site are: - <A HREF=https://www.algebra.com/algebra/homework/Systems-of-equations/Solving-the-system-of-alg-eqns-of-deg2-deg1.lesson>Solving systems of algebraic equations of degree 2 and degree 1</A> - <A HREF=http://www.algebra.com/algebra/homework/Systems-of-equations/Solving-the-system-of-algebraic-equations-of-degree-2.lesson>Solving systems of algebraic equations of degree 2</A> - <A HREF=https://www.algebra.com/algebra/homework/Systems-of-equations/Solving-typical-problems-on-systems-of-non-linear-equations-from-the-archive.lesson>Solving typical problems on systems of non-linear equations</A> - <A HREF=https://www.algebra.com/algebra/homework/Systems-of-equations/Some-tricks-to-solve-systems-of-non-linear-alg-eqns.lesson>Some tricks to solve systems of non-linear algebraic equations</A> - <A HREF=https://www.algebra.com/algebra/homework/Systems-of-equations/Geometric-word-problems-that-are-solved-using-systems-of-non-linear-equations.lesson>Geometric word problems that are solved using systems of non-linear equations</A> - <A HREF=https://www.algebra.com/algebra/homework/Systems-of-equations/Math-circle-level-problem-on-solving-a-system-of-non-linear-equations.lesson>Math circle level problems on solving systems of non-linear equations</A> - <A HREF=https://www.algebra.com/algebra/homework/Systems-of-equations/Solving-some-special-systems-of-non-linear-algebraic-equations.lesson>Solving some special systems of non-linear algebraic equations</A> - <A HREF=https://www.algebra.com/algebra/homework/Systems-of-equations/The-system-of-nonlinear-alg-equations-with-symm-functions-in-the-left-side.lesson>Solving systems of non-linear algebraic equations with symmetric functions of unknowns</A> - <A HREF=https://www.algebra.com/algebra/homework/Systems-of-equations/OVERVIEW-of-lessons-on-solving-non-linear-equations-and-systems.lesson>OVERVIEW of lessons on solving systems of non-linear equations in two or more unknowns</A> My other lessons on solving systems of non-linear equations in this site are - <A HREF=http://www.algebra.com/algebra/homework/Matrices-and-determiminant/Solving-systems-of-non-linear-equations-in-two-unknowns-using-Cramer%27s-rule.lesson>Solving systems of non-linear equations in two unknowns using the Cramer's rule</A> - <A HREF=http://www.algebra.com/algebra/homework/Matrices-and-determiminant/Solving-systems-of-non-linear-equations-in-three-unknowns-using-Cramer%27s-rule.lesson>Solving systems of non-linear equations in three unknowns using Cramer's rule</A> Use this file/link <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-I - YOUR ONLINE TEXTBOOK</A> to navigate over all topics and lessons of the online textbook ALGEBRA-I.
