Question

What are the zeros of f(x)=−4(2x+3)3(6−x) ? (1 point) Responses x=−32,6 x equals negative Start Fraction 3 over 2 End Fraction comma 6 x=−32,6,0 x equals negative Start Fraction 3 over 2 End Fraction comma 6 comma 0 x=−32,6,−4 x equals negative Start Fraction 3 over 2 End Fraction comma 6 comma negative 4 x=32,−6 x equals Start Fraction 3 over 2 End Fraction comma negative 6

Answer 1

To find the zeros of the function \( f(x) = -4(2x + 3)^3(6 - x) \), we need to set the function equal to zero:

\[
-4(2x + 3)^3(6 - x) = 0
\]

This product is equal to zero when either \( (2x + 3)^3 = 0 \) or \( (6 - x) = 0 \).

1. **Solving \( (2x + 3)^3 = 0 \)**:

Taking the cube root, we have:
\[
2x + 3 = 0
\]
Solving for \( x \):
\[
2x = -3 \implies x = -\frac{3}{2}
\]

2. **Solving \( (6 - x) = 0 \)**:

Rearranging gives:
\[
x = 6
\]

Now, we have found the zeros:

- From the first equation, \( x = -\frac{3}{2} \).
- From the second equation, \( x = 6 \).

So the zeros of \( f(x) \) are:
\[
x = -\frac{3}{2}, 6
\]

Thus, the correct response is:

**x equals negative Start Fraction 3 over 2 End Fraction, comma 6**.