Question

if $f(x) = x^3 - 3x^2 - 13x + 15$, which of the following is not a factor of $f(x)$?

answer

$\circ$ $(x - 1)$
$\circ$ $(x + 3)$
$\circ$ $(x - 5)$
$\circ$ $(x + 7)$

Explanation:

Step1: Recall Factor Theorem

The Factor Theorem states that if \((x - a)\) is a factor of \(f(x)\), then \(f(a)=0\). For a factor \((x + b)=(x - (-b))\), we check \(f(-b)=0\).

Step2: Check \((x - 1)\)

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

$$
LATEXBLOCK0
$$

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

Step3: Check \((x + 3)\)

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

$$
LATEXBLOCK1
$$

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

Step4: Check \((x - 5)\)

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

$$
LATEXBLOCK2
$$

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

Step5: Check \((x + 7)\)

For \((x + 7)=(x - (-7))\), \(a=-7\). Calculate \(f(-7)\):

$$
LATEXBLOCK3
$$

So \((x + 7)\) is not a factor.

Answer:

\((x + 7)\)