Skill: Use the factor theorem

For a polynomial $(f(x))$, the factor theorem says

$$x-a \text{ is a factor of } f(x) \quad \Longleftrightarrow \quad f(a)=0.$$

So, to check whether $(x-a)$ is a factor:

1. Identify $a$.
2. Substitute $a$ into $(f(x))$.
3. If $(f(a)=0)$, then $(x-a)$ is a factor.
4. If $(f(a)\neq 0)$, then $(x-a)$ is not a factor.

Example

Determine whether $(x-2)$ is a factor of

$$f(x)=x^3-4x^2+x+6.$$

Since the possible factor is $(x-2)$, use $(a=2)$. Evaluate $(f(2))$:

$$\begin{aligned} f(2) &=2^3-4(2^2)+2+6\\ &=8-16+2+6\\ &=0. \end{aligned}$$

Because $(f(2)=0)$, the factor theorem tells us that

$$x-2 \text{ is a factor of } f(x).$$

Dividing $(f(x))$ by $(x-2)$ gives

$$f(x)=(x-2)(x^2-2x-3).$$

The quadratic factors further:

$$x^2-2x-3=(x-3)(x+1).$$

Therefore,

$$\boxed{f(x)=(x-2)(x-3)(x+1)}.$$

Also, $(f(2)=0)$ means that $(x=2)$ is a zero of the polynomial, so the graph of $(y=f(x))$ crosses or touches the $x$-axis at $((2,0))$.