Factor the expression completely or state that the polynomial is prime: 9x^3 - 9x.

Step-by-step solution

1. 1

   Take out the full greatest common factor. Both terms share the number $9$ and also the variable $x$, so the GCF is $9x$, not merely $9$. Missing the variable part is the usual reason this problem is left half-factored: $9x^{3}-9x=9x\left(x^{2}-1\right).$

2. 2

   Identify the leftover pattern. The bracket $x^{2}-1$ is a difference of two squares, since $1=1^{2}$.

3. 3

   Apply the difference-of-squares identity. With $a=x$ and $b=1$ in $a^{2}-b^{2}=(a-b)(a+b)$, $x^{2}-1=(x-1)(x+1).$

4. 4

   Combine into the complete factorization. $9x^{3}-9x=9x(x-1)(x+1).$ All three factors are linear, so no further factoring is possible.

5. 5

   Verify by expanding and by roots. Expanding: $9x\left(x^{2}-1\right)=9x^{3}-9x$. As a check on the factors, the roots of the original are $x=0,\pm 1$, exactly the values that make $9x$, $x-1$ and $x+1$ vanish.