Question

Find the following product:

(x² − 1) (x⁴ + x² + 1)

Answer 1

 Given ${\displaystyle (x^2 -1)(x^4+x^2+1)}$

We shall use the identity  ${\displaystyle (a-b)(a^2+ab+b^2)=a^3-b^3}$

We can rearrange the ${\displaystyle (x^2-1)(x^4+x^2+1)}$as

$$(x^2-1)[{\left(x^2\right)}^2+\left(x^2\right)\left(1\right)+{\left(1\right)}^2]$$

${\displaystyle ={\left(x^2\right)}^3-{\left(1\right)}^3}$

${\displaystyle =\left(x^2\right)\times \left(x^2\right)\times \left(x^2\right)-\left(1\right)\times \left(1\right)\times \left(1\right)}$

${\displaystyle =x^6-1^3}$

${\displaystyle =x^6-1}$

Hence the Product value of ${\displaystyle (x^2-1)(x^4+x^2+1)}$is  ${\displaystyle x^6-1}$.