Question

2 determine if d(x) is a factor of f(x)
a) $f(x) = x^3 - 5x^2 - 7x - 2$
$d(x) = x + 1$
b)

Explanation:

Step1: Recall Factor Theorem

The Factor Theorem states that a polynomial \( d(x) = x - c \) is a factor of \( f(x) \) if and only if \( f(c) = 0 \). For \( d(x)=x + 1=x-(-1) \), we need to check if \( f(-1)=0 \).

Step2: Substitute \( x=-1 \) into \( f(x) \)

Given \( f(x)=x^{3}-5x^{2}-7x - 2 \), substitute \( x = - 1 \):

$$
LATEXBLOCK0
$$

Step3: Analyze the result

Since \( f(-1)=-1
eq0 \), by the Factor Theorem, \( d(x)=x + 1 \) is not a factor of \( f(x)=x^{3}-5x^{2}-7x - 2 \).

Answer:

\( d(x)=x + 1 \) is not a factor of \( f(x)=x^{3}-5x^{2}-7x - 2 \)