Question
What are the zeros of f(x)=-5(4x+9)^2(8-x)
?
A
x=−94, 8, −5x=-\frac{9}{4},\ 8,\ -5x=−
4
9
, 8, −5
B
x=−94, 8, 0x=-\frac{9}{4},\ 8,\ 0x=−
4
9
, 8, 0
C
x=−94, 8x=-\frac{9}{4},\ 8x=−
4
9
, 8
D
x=94, −8x=\frac{9}{4},\ -8x=
4
9
, −8
Answers
Answered by
GPT-4o mini
To find the zeros of the function \( f(x) = -5(4x + 9)^2(8 - x) \), we set \( f(x) = 0 \):
\[
-5(4x + 9)^2(8 - x) = 0
\]
Since the product of terms equals zero, we can set each factor to zero. This gives us:
1. \( -5 \neq 0 \) (not a zero)
2. \( (4x + 9)^2 = 0 \)
3. \( (8 - x) = 0 \)
Now, solve for each case:
For \( (4x + 9)^2 = 0 \):
\[
4x + 9 = 0
\]
\[
4x = -9
\]
\[
x = -\frac{9}{4}
\]
For \( (8 - x) = 0 \):
\[
x = 8
\]
Now we summarize the zeros:
1. \( x = -\frac{9}{4} \)
2. \( x = 8 \)
Thus, the zeros of the function are \( x = -\frac{9}{4} \) and \( x = 8 \).
Looking at the options you provided:
- A, B, and C include \( -\frac{9}{4} \) and \( 8 \).
- D does not match our solution.
The correct representation would therefore be:
\( x = -\frac{9}{4}, 8 \)
However, the formats in the options are confusing and potentially have errors. If we interpret them properly, the correct option appears to include these values. Based on what you provided, option **C** is the most accurate if it lists those zeros without extraneous information. But please verify the formatting issue you're encountering, as they seem incorrectly transcribed.
If you need further clarification, please provide the intended formatting or context.
\[
-5(4x + 9)^2(8 - x) = 0
\]
Since the product of terms equals zero, we can set each factor to zero. This gives us:
1. \( -5 \neq 0 \) (not a zero)
2. \( (4x + 9)^2 = 0 \)
3. \( (8 - x) = 0 \)
Now, solve for each case:
For \( (4x + 9)^2 = 0 \):
\[
4x + 9 = 0
\]
\[
4x = -9
\]
\[
x = -\frac{9}{4}
\]
For \( (8 - x) = 0 \):
\[
x = 8
\]
Now we summarize the zeros:
1. \( x = -\frac{9}{4} \)
2. \( x = 8 \)
Thus, the zeros of the function are \( x = -\frac{9}{4} \) and \( x = 8 \).
Looking at the options you provided:
- A, B, and C include \( -\frac{9}{4} \) and \( 8 \).
- D does not match our solution.
The correct representation would therefore be:
\( x = -\frac{9}{4}, 8 \)
However, the formats in the options are confusing and potentially have errors. If we interpret them properly, the correct option appears to include these values. Based on what you provided, option **C** is the most accurate if it lists those zeros without extraneous information. But please verify the formatting issue you're encountering, as they seem incorrectly transcribed.
If you need further clarification, please provide the intended formatting or context.