Certainly! Let's find the zeros of the function [tex]\( g(x) = x^3 + 6x^2 - 9x - 54 \)[/tex]. The zeros of a function are the values of [tex]\( x \)[/tex] for which the function equals zero, i.e., [tex]\( g(x) = 0 \)[/tex].
Here's the step-by-step solution:
1. Identify the function:
[tex]\[ g(x) = x^3 + 6x^2 - 9x - 54 \][/tex]
2. Solve the equation [tex]\( g(x) = 0 \)[/tex]:
We need to find the values of [tex]\( x \)[/tex] that satisfy:
[tex]\[ x^3 + 6x^2 - 9x - 54 = 0 \][/tex]
3. Perform the factorization (if possible) or use other algebraic methods to find the roots.
4. List all possible real roots and test them:
For this cubic polynomial, one common method is to use the Rational Root Theorem, which suggests that any rational root of the polynomial [tex]\( P(x) = 0 \)[/tex] is a factor of the constant term (-54) divided by a factor of the leading coefficient (1). Therefore, we test possible integral roots:
[tex]\[ \pm 1, \pm 2, \pm 3, \pm 6, \pm 9, \pm 18, \pm 27, \pm 54 \][/tex]
After verifying, we determine that the roots are [tex]\( 3, -3, -6 \)[/tex].
5. Compare the determined zeros with the given choices:
- Choice A: [tex]\( 1, 2, 27 \)[/tex]
- Choice B: [tex]\( 3, -3, -6 \)[/tex]
- Choice C: [tex]\( -6, 3, 6 \)[/tex]
- Choice D: [tex]\( 2, -1, -18 \)[/tex]
6. Conclusion:
The zeros of the polynomial [tex]\( g(x) = x^3 + 6x^2 - 9x - 54 \)[/tex] are [tex]\( 3, -3, -6 \)[/tex].
Thus, the correct answer is:
[tex]\[ \text{B. } 3, -3, -6 \][/tex]
