Lesson OVERVIEW of Factoring the binomials x^n-a^n and x^n+a^n
Algebra
->
Polynomials-and-rational-expressions
-> Lesson OVERVIEW of Factoring the binomials x^n-a^n and x^n+a^n
Log On
Algebra: Polynomials, rational expressions and equations
Section
Solvers
Solvers
Lessons
Lessons
Answers archive
Answers
Source code of 'OVERVIEW of Factoring the binomials x^n-a^n and x^n+a^n'
This Lesson (OVERVIEW of Factoring the binomials x^n-a^n and x^n+a^n)
was created by by
ikleyn(52776)
:
View Source
,
Show
About ikleyn
:
<H2>Overview of Factoring the binomials {{{x^n-a^n}}} and {{{x^n+a^n}}}</H2> This lesson is the overview of factoring the binomials {{{x^n-a^n}}} and {{{x^n+a^n}}}. It is based on the two preceding lessons <A HREF=http://www.algebra.com/algebra/homework/Polynomials-and-rational-expressions/Divisibility-of-the-Binomial-%28x%5En-a%5En%29-by-%28x-a%29.lesson>Factoring the binomials {{{x^n-a^n}}}</A> and <A HREF=http://www.algebra.com/algebra/homework/Polynomials-and-rational-expressions/Factoring-the-binomials-x%5En+a%5En-for-odd-degrees.lesson>Factoring the binomials {{{x^n+a^n}}} for odd degrees</A> that are under the current topic in this site. For all the proofs, examples, motivations and other details consult these two lessons. <H3>Factoring the binomials {{{x^n-a^n}}}</H3> 1. For all real numbers {{{a}}} and {{{b}}} and for any integer index {{{n}}} greater than or equal to 2 the formula is valid {{{a^n -b^n}}} = {{{(a-b)*(a^(n-1) + a^(n-2)*b + a^(n-3)*b^2 + ellipsis + a*b^(n-2) + b^(n-1))}}}. 2. For any integer index {{{n}}} greater than or equal to 2 and for any real number {{{a}}} the binomial {{{x^n -a^n}}} is divided by the linear binomial {{{x -a}}}. The formula is valid {{{x^n -a^n}}} = {{{(x-a)*(x^(n-1) + a*x^(n-2) + a^2*x^(n-3) + ellipsis + a^(n-2)*x + a^(n-1))}}}. This formula is factoring the binomial {{{x^n -a^n}}}   into the product of the linear binomial {{{x -a}}} and the polynomial {{{x^(n-1) + a*x^(n-2) + a^2*x^(n-3) + ellipsis + a^(n-2)*x + a^(n-1))}}}. The quotient of division the binomial {{{x^n -a^n}}}   by the binomial {{{x -a}}}   is the polynomial {{{x^(n-1) + a*x^(n-2) + a^2*x^(n-3) + ellipsis + a^(n-2)*x + a^(n-1))}}}. 3. For any integer index {{{n}}} greater than or equal to 2 the binomial {{{x^n-1}}} is divided by the linear binomial {{{x-1}}}. The formula is valid {{{x^n-1}}} = {{{(x-1)*(x^(n-1) + x^(n-2) + x^(n-3) + ellipsis + x + 1)}}}. This formula is factoring the binomial {{{x^n-1}}}   into the product of the linear binomial {{{x-1}}} and the polynomial {{{x^(n-1) + x^(n-2) + x^(n-3) + ellipsis + x + 1)}}}. The quotient of division the binomial {{{x^n-1}}}   by the binomial {{{x-1}}}   is the polynomial {{{x^(n-1) + x^(n-2) + x^(n-3) + ellipsis + x + 1)}}}: {{{(x^n -1)/(x-1)}}} = {{{x^(n-1) + x^(n-2) + x^(n-3) + ellipsis + x + 1}}}. 4. The sum of the first {{{n}}} terms of the geometric progression {{{a}}}, {{{a*q}}}, {{{a*q^2}}}, ..., {{{a*q^(n-1)}}} is equal to {{{S[n]}}} = {{{a + a*q + a*q^2 + ellipsis + aq^(n-1)}}} = {{{(a*q^n - a)/(q-1)}}}. <H3>Factoring the binomials {{{x^n+a^n}}}</H3> 1. The formula is valid {{{a^n +b^n}}} = {{{(a+b)*(a^(n-1) - a^(n-2)*b + a^(n-3)*b^2 + ellipsis - a*b^(n-2) + b^(n-1))}}} for all real numbers {{{a}}} and {{{b}}} and for odd integer index {{{n}}} greater than or equal to 3. 2. For odd integer index {{{n}}} greater than or equal to 3 and for any real number {{{a}}} the binomial {{{x^n +a^n}}} is divided by the linear binomial {{{x +a}}}. The formula is valid {{{x^n +a^n}}} = {{{(x+a)*(x^(n-1) - a*x^(n-2) + a^2*x^(n-3) - ellipsis - a^(n-2)*x + a^(n-1))}}}. This formula is factoring the binomial {{{x^n +a^n}}}   into the product of the linear binomial {{{x +a}}} and the polynomial {{{x^(n-1) - a*x^(n-2) + a^2*x^(n-3) - ellipsis - a^(n-2)*x + a^(n-1))}}}. The quotient of division the binomial {{{x^n +a^n}}}   by the binomial {{{x +a}}}   is the polynomial {{{x^(n-1) - a*x^(n-2) + a^2*x^(n-3) - ellipsis - a^(n-2)*x + a^(n-1))}}}. 3. For odd integer index {{{n}}} greater than or equal to 3 the binomial {{{x^n+1}}} is divided by the linear binomial {{{x+1}}}. The formula is valid {{{x^n+1}}} = {{{(x+1)*(x^(n-1) - x^(n-2) + x^(n-3) + ellipsis - x + 1)}}}. This formula is factoring the binomial {{{x^n+1}}}   into the product of the linear binomial {{{x+1}}} and the polynomial {{{x^(n-1) - x^(n-2) + x^(n-3) - ellipsis - x + 1)}}}. The quotient of division the binomial {{{x^n+1}}}   by the binomial {{{x+1}}}   is the polynomial {{{x^(n-1) - x^(n-2) + x^(n-3) - ellipsis - x + 1)}}}: {{{(x^n +1)/(x+1)}}} = {{{x^(n-1) - x^(n-2) + x^(n-3) - ellipsis - x + 1}}}. 4. The binomial {{{x^n+a^n}}} is not divisible by the binomial {{{x+a}}} for even integer index {{{n}}}.
