Determine if Prime x^3-6x^2+12x-8

If a polynomial function has integer coefficients, then every rational zero will have the form ${\displaystyle \frac{p}{q}}$ where ${\displaystyle p}$ is a factor of the constant and ${\displaystyle q}$ is a factor of the leading coefficient.

Find every combination of ${\displaystyle \pm \frac{p}{q}}$. These are the possible roots of the polynomial function.

Substitute ${\displaystyle 2}$ and simplify the expression. In this case, the expression is equal to ${\displaystyle 0}$ so ${\displaystyle 2}$ is a root of the polynomial.

Since ${\displaystyle 2}$ is a known root, divide the polynomial by ${\displaystyle x-2}$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

Divide ${\displaystyle x^3-6x^2+12x-8}$ by ${\displaystyle x-2}$.

Write ${\displaystyle x^3-6x^2+12x-8}$ as a set of factors.

Rewrite ${\displaystyle 4}$ as ${\displaystyle 2^2}$.

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

Rewrite the polynomial.

Factor using the perfect square trinomial rule ${\displaystyle a^2-2ab+b^2={(a-b)}^2}$, where ${\displaystyle a=x}$ and ${\displaystyle b=2}$.

Raise ${\displaystyle x-2}$ to the power of ${\displaystyle 1}$.

Use the power rule ${\displaystyle a^m{a}^n=a^{m+n}}$ to combine exponents.

Add ${\displaystyle 1}$ and ${\displaystyle 2}$.