Answer :
To solve for the zeros of the function [tex]\( h(x) = (x^2 - 49)(x + 4) \)[/tex], we follow these steps:
1. Set the function equal to zero:
[tex]\[ (x^2 - 49)(x + 4) = 0 \][/tex]
2. Factorize [tex]\( x^2 - 49 \)[/tex]:
We recognize that [tex]\( x^2 - 49 \)[/tex] is a difference of squares, which can be factored as:
[tex]\[ x^2 - 49 = (x - 7)(x + 7) \][/tex]
3. Substitute the factored form back into the equation:
[tex]\[ (x - 7)(x + 7)(x + 4) = 0 \][/tex]
4. Find the roots by setting each factor equal to zero:
[tex]\[ x - 7 = 0 \quad \Rightarrow \quad x = 7 \][/tex]
[tex]\[ x + 7 = 0 \quad \Rightarrow \quad x = -7 \][/tex]
[tex]\[ x + 4 = 0 \quad \Rightarrow \quad x = -4 \][/tex]
Thus, the zeros of the function are [tex]\( x = 7 \)[/tex], [tex]\( x = -7 \)[/tex], and [tex]\( x = -4 \)[/tex].
Among the provided answer choices:
- A. 0 is not a zero of [tex]\( h(x) \)[/tex].
- B. 4 is not a zero of [tex]\( h(x) \)[/tex].
- C. 7 is a zero of [tex]\( h(x) \)[/tex].
- D. 49 is not a zero of [tex]\( h(x) \)[/tex].
Therefore, the correct answer is:
C. 7
1. Set the function equal to zero:
[tex]\[ (x^2 - 49)(x + 4) = 0 \][/tex]
2. Factorize [tex]\( x^2 - 49 \)[/tex]:
We recognize that [tex]\( x^2 - 49 \)[/tex] is a difference of squares, which can be factored as:
[tex]\[ x^2 - 49 = (x - 7)(x + 7) \][/tex]
3. Substitute the factored form back into the equation:
[tex]\[ (x - 7)(x + 7)(x + 4) = 0 \][/tex]
4. Find the roots by setting each factor equal to zero:
[tex]\[ x - 7 = 0 \quad \Rightarrow \quad x = 7 \][/tex]
[tex]\[ x + 7 = 0 \quad \Rightarrow \quad x = -7 \][/tex]
[tex]\[ x + 4 = 0 \quad \Rightarrow \quad x = -4 \][/tex]
Thus, the zeros of the function are [tex]\( x = 7 \)[/tex], [tex]\( x = -7 \)[/tex], and [tex]\( x = -4 \)[/tex].
Among the provided answer choices:
- A. 0 is not a zero of [tex]\( h(x) \)[/tex].
- B. 4 is not a zero of [tex]\( h(x) \)[/tex].
- C. 7 is a zero of [tex]\( h(x) \)[/tex].
- D. 49 is not a zero of [tex]\( h(x) \)[/tex].
Therefore, the correct answer is:
C. 7
