To determine the correct factors of the polynomial function [tex]\( g(x) = x^3 + 2x^2 - x - 2 \)[/tex], we need to factorize the polynomial. Here's a detailed, step-by-step guide to finding the factors:
1. Identify the Polynomial: The given polynomial is [tex]\( g(x) = x^3 + 2x^2 - x - 2 \)[/tex].
2. Check for Rational Roots: According to the Rational Root Theorem, the possible rational roots of a polynomial are the factors of the constant term divided by the factors of the leading coefficient. In this case, the constant term is [tex]\(-2\)[/tex] and the leading coefficient is [tex]\(1\)[/tex].
Therefore, the possible rational roots are:
[tex]\[
\pm 1, \pm 2
\][/tex]
3. Test the Possible Roots:
- Check [tex]\( x = 1 \)[/tex]:
[tex]\[
g(1) = 1^3 + 2(1)^2 - 1 - 2 = 1 + 2 - 1 - 2 = 0
\][/tex]
Since [tex]\( g(1) = 0 \)[/tex], [tex]\( x = 1 \)[/tex] is a root. This means [tex]\( x - 1 \)[/tex] is a factor.
- Check [tex]\( x = -1 \)[/tex]:
[tex]\[
g(-1) = (-1)^3 + 2(-1)^2 - (-1) - 2 = -1 + 2 + 1 - 2 = 0
\][/tex]
Since [tex]\( g(-1) = 0 \)[/tex], [tex]\( x = -1 \)[/tex] is also a root. This means [tex]\( x + 1 \)[/tex] is a factor.
- Check [tex]\( x = 2 \)[/tex]:
[tex]\[
g(2) = 2^3 + 2(2)^2 - 2 - 2 = 8 + 8 - 2 - 2 = 12
\][/tex]
Since [tex]\( g(2) \neq 0 \)[/tex], [tex]\( x = 2 \)[/tex] is not a root.
- Check [tex]\( x = -2 \)[/tex]:
[tex]\[
g(-2) = (-2)^3 + 2(-2)^2 - (-2) - 2 = -8 + 8 + 2 - 2 = 0
\][/tex]
Since [tex]\( g(-2) = 0 \)[/tex], [tex]\( x = -2 \)[/tex] is a root. This means [tex]\( x + 2 \)[/tex] is a factor.
4. List the Factors: From our root tests, the factors of [tex]\( g(x) = x^3 + 2x^2 - x - 2 \)[/tex] are:
[tex]\[
(x - 1), (x + 1), (x + 2)
\][/tex]
5. Compare with Given Options:
- A. [tex]\( (x - 1), (x + 1) \)[/tex]
- B. [tex]\( (x - 2), (x - 1), (x + 1) \)[/tex]
- C. [tex]\( (x - 2), (x - 1), (x + 2) \)[/tex]
- D. [tex]\( (x - 1), (x + 1), (x + 2) \)[/tex]
Since the correct factors are [tex]\( (x - 1), (x + 1), (x + 2) \)[/tex], the correct answer is:
[tex]\[
\boxed{\text{D}}
\][/tex]
