To determine which expression is a factor of the polynomial [tex]\(x^4 - x^3 - 11x^2 + 5x + 30\)[/tex], we can utilize the Factor Theorem. The Factor Theorem states that [tex]\(x-c\)[/tex] is a factor of a polynomial if and only if the value of the polynomial evaluated at [tex]\(c\)[/tex] is zero, i.e., [tex]\(P(c) = 0\)[/tex].
Let's denote the polynomial by [tex]\(P(x) = x^4 - x^3 - 11x^2 + 5x + 30\)[/tex].
We'll check each potential factor by evaluating the polynomial at specific values:
1. For [tex]\(x + 2\)[/tex], check if [tex]\(P(-2) = 0\)[/tex]:
[tex]\[
P(-2) = (-2)^4 - (-2)^3 - 11(-2)^2 + 5(-2) + 30
\][/tex]
[tex]\[
= 16 + 8 - 44 - 10 + 30
\][/tex]
[tex]\[
= 16 + 8 - 44 - 10 + 30
\][/tex]
[tex]\[
= 54 - 54
\][/tex]
[tex]\[
= 0
\][/tex]
Since [tex]\(P(-2) = 0\)[/tex], [tex]\(x + 2\)[/tex] is a factor of the polynomial.
2. For [tex]\(x - 2\)[/tex], check if [tex]\(P(2) = 0\)[/tex]:
[tex]\[
P(2) = 2^4 - 2^3 - 11(2)^2 + 5(2) + 30
\][/tex]
[tex]\[
= 16 - 8 - 44 + 10 + 30
\][/tex]
[tex]\[
= 48 - 52
\][/tex]
[tex]\[
= -4
\][/tex]
Since [tex]\(P(2) \neq 0\)[/tex], [tex]\(x - 2\)[/tex] is not a factor.
3. For [tex]\(x + 5\)[/tex], check if [tex]\(P(-5) = 0\)[/tex]:
[tex]\[
P(-5) = (-5)^4 - (-5)^3 - 11(-5)^2 + 5(-5) + 30
\][/tex]
[tex]\[
= 625 + 125 - 275 - 25 + 30
\][/tex]
[tex]\[
= 780 - 300
\][/tex]
[tex]\[
= 480
\][/tex]
Since [tex]\(P(-5) \neq 0\)[/tex], [tex]\(x + 5\)[/tex] is not a factor.
4. For [tex]\(x - 5\)[/tex], check if [tex]\(P(5) = 0\)[/tex]:
[tex]\[
P(5) = 5^4 - 5^3 - 11(5)^2 + 5(5) + 30
\][/tex]
[tex]\[
= 625 - 125 - 275 + 25 + 30
\][/tex]
[tex]\[
= 655 - 400
\][/tex]
[tex]\[
= 255
\][/tex]
Since [tex]\(P(5) \neq 0\)[/tex], [tex]\(x - 5\)[/tex] is not a factor.
Based on our evaluations, the correct factor of [tex]\(x^4 - x^3 - 11x^2 + 5x + 30\)[/tex] is:
(1) [tex]\(x + 2\)[/tex]
