Question

which of the following functions has a factor of $x + 2$?
answer
$\circ\\ f(x) = x^3 - 14x^2 + 59x - 70$
$\circ\\ f(x) = x^3 - 14x^2 + 59x - 72$
$\circ\\ f(x) = x^3 - 10x^2 + 11x + 68$
$\circ\\ f(x) = x^3 - 10x^2 + 11x + 70$

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 \( x + 2 \), we have \( a = 2 \), so we need to check \( f(-2) = 0 \) for each function.

Step2: Evaluate \( f(-2) \) for the first function

For \( f(x) = x^3 - 14x^2 + 59x - 70 \):

$$
LATEXBLOCK0
$$

Step3: Evaluate \( f(-2) \) for the second function

For \( f(x) = x^3 - 14x^2 + 59x - 72 \):

$$
LATEXBLOCK1
$$

Step4: Evaluate \( f(-2) \) for the third function

For \( f(x) = x^3 - 10x^2 + 11x + 68 \):

$$
LATEXBLOCK2
$$

Step5: Evaluate \( f(-2) \) for the fourth function

For \( f(x) = x^3 - 10x^2 + 11x + 70 \):

$$
LATEXBLOCK3
$$

Answer:

\( f(x) = x^3 - 10x^2 + 11x + 70 \) (the fourth option)