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:

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 a factor of the form \((x + b)\), this is equivalent to \((x - (-b))\), so we check \(f(-b)=0\).

Step2: Check \((x + 3)\)

For \((x + 3)\), \(a=-3\). Calculate \(f(-3)\):

$$
LATEXBLOCK0
$$

So \((x + 3)\) is a factor.

Step3: Check \((x - 9)\)

For \((x - 9)\), \(a = 9\). Calculate \(f(9)\):

$$
LATEXBLOCK1
$$

Step4: Check \((x + 1)\) (for completeness)

For \((x + 1)\), \(a=-1\). Calculate \(f(-1)\):

$$
LATEXBLOCK2
$$

So \((x + 1)\) is a factor.

Step5: Check \((x + 5)\) (for completeness)

For \((x + 5)\), \(a=-5\). Calculate \(f(-5)\):

$$
LATEXBLOCK3
$$

So \((x + 5)\) is a factor.

Answer:

B. \((x - 9)\)