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

Question

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(1 point)
Responses

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=−32,6,0
x equals negative Start Fraction 3 over 2 End Fraction comma 6 comma 0

x=−32,6
x equals negative Start Fraction 3 over 2 End Fraction comma 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 equation will be zero if either \( (2x + 3)^3 = 0 \) or \( (6 - x) = 0 \).

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

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

Putting it all together, the zeros of the function are:

\[ x = -\frac{3}{2}, \quad x = 6 \]

Now for the options provided, we can match them:

• The correct response is: \[ x = -\frac{3}{2}, ; 6, ; 0 \]

However, 0 is not a zero of the function, so this option is incorrect.

The correct zeroes found are \( x = -\frac{3}{2} \) and \( x = 6 \). The closest options given are:

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

This matches the zeros found above.