Question

determine if ((x + 3)) is a factor of (f(x) = x^3 + 2x^2 - 7x - 12)

Explanation:

Step1: Recall the Factor Theorem

The Factor Theorem states that if \((x - a)\) is a factor of a polynomial \(f(x)\), then \(f(a)=0\). For the factor \((x + 3)\), we can rewrite it as \((x - (-3))\), so \(a=-3\). We need to evaluate \(f(-3)\) for the polynomial \(f(x)=x^{3}+2x^{2}-7x - 12\).

Step2: Substitute \(x = - 3\) into \(f(x)\)

\[

$$\begin{align*} f(-3)&=(-3)^{3}+2(-3)^{2}-7(-3)-12\\ &=-27 + 2(9)+21 - 12\\ &=-27+18 + 21-12\\ &=(-27 - 12)+(18 + 21)\\ &=-39 + 39\\ &=0 \end{align*}$$

\]

Answer:

Since \(f(-3) = 0\) by the Factor Theorem, \((x + 3)\) is a factor of \(f(x)=x^{3}+2x^{2}-7x - 12\).