Question

Find the number of factors for the polynomial
f(x) = x9 - x.​

Answer 1

$f(x) = {x}^{9} - x$

$f(x) = x( {x}^{8} - {1}^{8} )$

Using,

${a}^{2} - {b}^{2} = (a + b)(a - b)$

We get,

$f(x) = x( {x}^{4} + 1)( {x}^{4} - 1)$

$f(x) = x( {x}^{4} + 1)( {x}^{2} + 1)( {x}^{2} - 1)$

$f(x) = x( {x}^{4} + 1)( {x}^{2} + 1)(x + 1)(x - 1)$

Hence, there are $5$ Factors of $f(x)$