Use the Factor Theorem to determine whether [tex]$x + 2$[/tex] is a factor of [tex]$P(x) = x^4 + x^3 - 3x - 14$[/tex]. Specifically, evaluate [tex][tex]$P$[/tex][/tex] at the proper value, and then determine whether [tex]$x + 2$[/tex] is a factor:

[tex]
P(\square) = \square
[/tex]

[tex]x + 2[/tex] is a factor of [tex]P(x)[/tex]

[tex]x + 2[/tex] is not a factor of [tex]P(x)[/tex]



Answer :

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]