To determine which number is a zero of the function [tex]\( h(x) = (x^2 - 49)(x + 4) \)[/tex], we need to find the values of [tex]\( x \)[/tex] for which [tex]\( h(x) = 0 \)[/tex].
1. Factor the polynomial [tex]\( x^2 - 49 \)[/tex]:
[tex]\[
x^2 - 49 \text{ can be written as a difference of squares: } (x - 7)(x + 7)
\][/tex]
Thus, the function [tex]\( h(x) \)[/tex] can be rewritten as:
[tex]\[
h(x) = (x - 7)(x + 7)(x + 4)
\][/tex]
2. Set each factor equal to zero to find the zeros:
[tex]\[
(x - 7) = 0
\][/tex]
Solving this equation:
[tex]\[
x = 7
\][/tex]
[tex]\[
(x + 7) = 0
\][/tex]
Solving this equation:
[tex]\[
x = -7
\][/tex]
[tex]\[
(x + 4) = 0
\][/tex]
Solving this equation:
[tex]\[
x = -4
\][/tex]
3. Identify the zeros:
The zeros of [tex]\( h(x) = (x - 7)(x + 7)(x + 4) \)[/tex] are [tex]\( x = 7 \)[/tex], [tex]\( x = -7 \)[/tex], and [tex]\( x = -4 \)[/tex].
4. Choose the correct answer from the provided options:
Among the options given:
- A. [tex]\( 0 \)[/tex] is not a zero
- B. [tex]\( 4 \)[/tex] is not a zero
- C. [tex]\( 7 \)[/tex] is a zero
- D. [tex]\( 49 \)[/tex] is not a zero
Therefore, the correct answer is:
[tex]\[
\boxed{7}
\][/tex]
