Use the factor theorem

For a polynomial $f(x)$ , the factor theorem establishes that $x-a$ is a factor exactly when $f(a)=0$ ; thus, evaluating a polynomial identifies whether $a$ is a zero and whether the corresponding linear factor can be used in its factorization. This reasoning connects algebraic factors with roots and graph intercepts and supports simplifying rational expressions, without extending to abstract polynomial rings or more advanced factorization over complex fields.

Detailed Explanation: 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:

Identify $a$ .
Substitute $a$ into $(f(x))$ .
If $(f(a)=0)$ , then $(x-a)$ is a factor.
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 $((2,0))$ .

Learn by doing: Use the factor theorem

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Factor Theorem - Is the Binomial a Factor (Yes/No)

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