Chapter 4.6, Problem 54WE

To determine

To Find:

To find the factor of $a^{6n}-b^{6n}$ assuming $n$ to be a positive integer.

Answer to Problem 54WE

The simplified factor of the polynomial $a^{6n}-b^{6n}$ is $\left(a^n-b^n\right)\left(a^{2n}+ab+b^{2n}\right)\left(a^n+b^n\right)\left(a^{2n}-ab+b^{2n}\right)$ .

Explanation of Solution

Given information:

The given polynomial is $a^{6n}-b^{6n}$ .

Formula:

Difference cube formula:

  $x^3-y^3=\left(x-y\right)\left(x^2+xy+y^2\right)$

Sum cube formula:

  $x^3+y^3=\left(x+y\right)\left(x^2-xy+y^2\right)$

To Factorize $a^{6n}-b^{6n}$ ,for $n>0$

  $\Rightarrow a^{6n}-b^{6n}=\left(a^{3n}-b^{3n}\right)\left(a^{3n}+b^{3n}\right)$

Apply the sum and difference cube formula: $x^3-y^3=\left(x-y\right)\left(x^2+xy+y^2\right)$ and $x^3+y^3=\left(x+y\right)\left(x^2-xy+y^2\right)$

  $=\left(a^n-b^n\right)\left(a^{2n}+ab+b^{2n}\right)\left(a^n+b^n\right)\left(a^{2n}-ab+b^{2n}\right)$

Therefore, the factor of the polynomial $a^{6n}-b^{6n}$ is $\left(a^n-b^n\right)\left(a^{2n}+ab+b^{2n}\right)\left(a^n+b^n\right)\left(a^{2n}-ab+b^{2n}\right)$ .