To determine whether [tex]\(x + 2\)[/tex] is a factor of [tex]\(P(x) = x^4 + x^3 - 3x - 14\)[/tex] using the Factor Theorem, we need to evaluate [tex]\(P\)[/tex] at [tex]\(x = -2\)[/tex]. According to the Factor Theorem, [tex]\(x + 2\)[/tex] is a factor of [tex]\(P(x)\)[/tex] if and only if [tex]\(P(-2) = 0\)[/tex].
Let's follow these steps:
1. Substitute [tex]\(x = -2\)[/tex] in [tex]\(P(x)\)[/tex]:
[tex]\[
P(-2) = (-2)^4 + (-2)^3 - 3(-2) - 14
\][/tex]
2. Evaluate each term:
[tex]\[
(-2)^4 = 16
\][/tex]
[tex]\[
(-2)^3 = -8
\][/tex]
[tex]\[
-3(-2) = 6
\][/tex]
[tex]\[
-14 \text{ (as it is)}
\][/tex]
3. Sum these values:
[tex]\[
P(-2) = 16 + (-8) + 6 - 14
\][/tex]
Simplify it step by step:
[tex]\[
P(-2) = 16 - 8 + 6 - 14
\][/tex]
[tex]\[
P(-2) = 8 + 6 - 14
\][/tex]
[tex]\[
P(-2) = 14 - 14
\][/tex]
[tex]\[
P(-2) = 0
\][/tex]
Since [tex]\(P(-2) = 0\)[/tex], we conclude:
[tex]\[
x+2 \text{ is a factor of } P(x)
\][/tex]
