To determine which function has the same zeros as [tex]\( g(x) = -x^3 + x^2 + 12x \)[/tex], let's go through the process of finding the zeros step-by-step.
1. Factor the Function:
The first step in finding the zeros of a function is often to factor it. Let's write [tex]\( g(x) \)[/tex] in its factored form:
[tex]\[
g(x) = -x^3 + x^2 + 12x
\][/tex]
Factor out the greatest common factor (GCF), which in this case is [tex]\( x \)[/tex]:
[tex]\[
g(x) = x(-x^2 + x + 12)
\][/tex]
2. Further Factor the Quadratic Expression:
Next, we need to factor the quadratic expression [tex]\(-x^2 + x + 12\)[/tex]. To do this, let's rewrite it in a standard form:
[tex]\[
-x^2 + x + 12
\][/tex]
One method is to look for two numbers that multiply to give the constant term (in this case, -12) and add up to the linear coefficient (in this case, 1).
The quadratic expression can be factored as:
[tex]\[
-x^2 + x + 12 = -(x^2 - x - 12)
\][/tex]
Now we look for factors of [tex]\(-12\)[/tex] that add to [tex]\(-1\)[/tex]:
[tex]\[
x^2 - x - 12 = (x - 4)(x + 3)
\][/tex]
3. Combine the Factored Terms:
Now, combining all the factored terms, we get:
[tex]\[
g(x) = x \cdot -(x - 4)(x + 3)
\][/tex]
4. Identify the Zeros:
A function is zero wherever any of its factors is zero. Therefore, set each factor to zero and solve for [tex]\( x \)[/tex]:
[tex]\[
x = 0, \quad -(x - 4) = 0, \quad \text{and} \quad (x + 3) = 0
\][/tex]
Solving these equations gives:
[tex]\[
x = 0, \quad x - 4 = 0 \implies x = 4, \quad \text{and} \quad x + 3 = 0 \implies x = -3
\][/tex]
5. Conclusion:
The zeros of the function [tex]\( g(x) = -x^3 + x^2 + 12x \)[/tex] are:
[tex]\[
x = -3, 0, \text{and} \ 4
\][/tex]
These are the values of [tex]\( x \)[/tex] where the function [tex]\( g(x) \)[/tex] equals zero.
Therefore, any function that has the same zeros as [tex]\( g(x) = -x^3 + x^2 + 12x \)[/tex] must also have zeros at [tex]\( x = -3 \)[/tex], [tex]\( x = 0 \)[/tex], and [tex]\( x = 4 \)[/tex].
