Question

Find the total number of factors of $(x^9-x)$.

• A. $5$
• B. $4$
• C. $2$
• D. Cannot be determined

Answer 1

The correct option is A $5$

$f\left(x\right)=x^9-x$
$=x(x^8-1)$
$=x[{\left(x^4\right)}^2-{\left(1\right)}^2]$
$=x\left[(x^4-1)(x^4+1)\right]$
$=x(x^4+1)[{\left(x^2\right)}^2-1^2]$
$=x(x^4+1)(x^2+1)(x^2-1)$
$=x(x^4+1)(x^2+1)(x-1)(x+1)$
$\therefore$ There are $5$ factors of $f\left(x\right)\Rightarrow \left(x\right),(x^4+1),(x^2+1),(x-1)$ and $(x+1)$