Question

if $f(x) = x^3 + 9x^2 + 23x + 15$, which of the following is not a factor of $f(x)$?

answer attempt 1 out of 2

$\circ$ $(x + 3)$
$\circ$ $(x - 9)$
$\circ$ $(x + 1)$
$\circ$ $(x + 5)$
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Explanation:

To determine which of the given options is not a factor of \( f(x) = x^3 + 9x^2 + 23x + 15 \), we can use the Factor Theorem. The Factor Theorem states that if \( (x - a) \) is a factor of a polynomial \( f(x) \), then \( f(a) = 0 \). For a factor of the form \( (x + b) \), this is equivalent to \( f(-b) = 0 \).

Step 1: Test \( (x + 3) \) (i.e., \( a = -3 \))

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

$$
LATEXBLOCK0
$$

Since \( f(-3) = 0 \), \( (x + 3) \) is a factor.

Step 2: Test \( (x - 9) \) (i.e., \( a = 9 \))

Substitute \( x = 9 \) into \( f(x) \):

$$
LATEXBLOCK1
$$

Since \( f(9) = 1680
eq 0 \), \( (x - 9) \) is not a factor. (We can stop here, but for completeness, we'll check the other options.)

Step 3: Test \( (x + 1) \) (i.e., \( a = -1 \))

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

$$
LATEXBLOCK2
$$

Since \( f(-1) = 0 \), \( (x + 1) \) is a factor.

Step 4: Test \( (x + 5) \) (i.e., \( a = -5 \))

Substitute \( x = -5 \) into \( f(x) \):

$$
LATEXBLOCK3
$$

Since \( f(-5) = 0 \), \( (x + 5) \) is a factor.

Answer:

B. \( (x - 9) \)