Question

How many factors does the expression $x^9-x$ have?

Answer 1

$x^9-x=x(x^8-1)$

$=x\left[\right(x^4{)}^2-1]$

$=x\left[\right(x^4{)}^2-1^2]=x(x^4-1\left)\right(x^4+1)$

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

$=x\left[\right(x^2{)}^2-1^2\left]\right(x^4+1)$

$=x(x^2-1)(x^2+1)(x^4+1)$

$=x(x-1)(x+1)(x^2+1)(x^4+1)$

Hence, the expression $x^9-x$ has 5 factors, namely,
$x,(x-1),(x+1),(x^2+1)$ and $(x^4+1)$