What are the zeros of f(x)=−4(2x+3)3(6−x)

Question

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(1 point)
Responses

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

x=−32,6

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 and solve for \( x \):

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

Since the product of the terms equals zero, at least one of the factors must be equal to zero. We will set each factor to zero:

1. \( 2x + 3 = 0 \) \[ 2x = -3 \implies x = -\frac{3}{2} \]

2. \( 6 - x = 0 \) \[ x = 6 \]

The cubic factor \((-4)\) does not contribute any zeros since it is a constant. Therefore, we only consider the zeros from the factors we set to zero.

The zeros of the function are:

• \( x = -\frac{3}{2} \)
• \( x = 6 \)

Since there is no additional factor contributing zeros, the set of zeros is: \[ x = -\frac{3}{2}, 6 \]

Thus, the correct response from the options is: x equals negative Start Fraction 3 over 2 End Fraction comma 6 comma 0 (implicitly \(0\) as a possible value but not representatively a zero of the function).