To find the real zeros of the function [tex]\( g(x) = x^3 + 2x^2 - x - 2 \)[/tex], follow these steps:
1. Understand the Problem: We need to determine the values of [tex]\( x \)[/tex] at which [tex]\( g(x) = 0 \)[/tex].
2. Set the Function to Zero:
[tex]\[ x^3 + 2x^2 - x - 2 = 0 \][/tex]
3. Solve the Equation: Identify the real roots of this polynomial equation by analyzing its factors, using methods such as factoring, synthetic division, or using a root-finding algorithm.
4. Verify the Roots:
Let's substitute the possible roots from the given choices into the equation to verify which ones satisfy [tex]\( g(x) = 0 \)[/tex].
- For [tex]\( x = 1 \)[/tex]:
[tex]\[ g(1) = 1^3 + 2(1)^2 - 1 - 2 = 1 + 2 - 1 - 2 = 0 \][/tex]
So, [tex]\( x = 1 \)[/tex] is a root.
- For [tex]\( x = -1 \)[/tex]:
[tex]\[ g(-1) = (-1)^3 + 2(-1)^2 - (-1) - 2 = -1 + 2 + 1 - 2 = 0 \][/tex]
So, [tex]\( x = -1 \)[/tex] is a root.
- For [tex]\( x = 2 \)[/tex]:
[tex]\[ g(2) = 2^3 + 2(2)^2 - 2 - 2 = 8 + 8 - 2 - 2 = 12 \][/tex]
[tex]\( g(2) \neq 0 \)[/tex], so [tex]\( x = 2 \)[/tex] is not a root.
- For [tex]\( x = -2 \)[/tex]:
[tex]\[ g(-2) = (-2)^3 + 2(-2)^2 - (-2) - 2 = -8 + 8 + 2 - 2 = 0 \][/tex]
So, [tex]\( x = -2 \)[/tex] is a root.
5. Check the Answer Choices:
Let's examine each of the given answer choices in light of our findings:
- A. [tex]\( 1, -1, 2 \)[/tex] (includes [tex]\( 2 \)[/tex], which is not a root)
- B. [tex]\( 1, -1, -2 \)[/tex] (all are roots, a correct match)
- C. [tex]\( 1, -1 \)[/tex] (misses [tex]\( -2 \)[/tex], so is incomplete)
- D. [tex]\( 2, -2, 1 \)[/tex] (includes [tex]\( 2 \)[/tex], which is not a root)
Hence, the correct answer is:
[tex]\[ \boxed{\text{B}} \][/tex]
