Chapter 6.7, Problem 69STP

To determine

To find:The factors of the given polynomial.

Answer to Problem 69STP

The factor of the given polynomial is $\left(a^2+b^2\right)\left(a^4-a^2{b}^2+b^4\right)$

Explanation of Solution

Given:

The polynomial is given as,

  $a^6+b^6$

Calculation:

Solve the given expression.

  $a^6+b^6={\left(a^2\right)}^3+{\left(b^2\right)}^3$

Use the below identity.

  $a^3+b^3=\left(a+b\right)\left(a^2-ab+b^2\right)$

Solve further as,

  $\begin{array}{c}{a}^6+b^6={\left(a^2\right)}^3+{\left(b^2\right)}^3\\ =\left(a^2+b^2\right)\left({\left(a^2\right)}^2-a^2{b}^2+{\left(b^2\right)}^2\right)\\ =\left(a^2+b^2\right)\left(a^4-a^2{b}^2+b^4\right)\end{array}$

Thus, the factor of the given polynomial is $\left(a^2+b^2\right)\left(a^4-a^2{b}^2+b^4\right)$