To determine which of the given numbers is a zero of the function [tex]\( g(x) = (x^2 - 36)(x + 7) \)[/tex], we need to find the values of [tex]\( x \)[/tex] for which [tex]\( g(x) = 0 \)[/tex].
Step-by-Step Solution:
1. Identify the polynomial factors:
The function [tex]\( g(x) \)[/tex] is given as [tex]\( g(x) = (x^2 - 36)(x + 7) \)[/tex].
2. Factor the polynomial expression:
We can factor [tex]\( x^2 - 36 \)[/tex] as a difference of squares:
[tex]\[
x^2 - 36 = (x - 6)(x + 6)
\][/tex]
Hence, we can rewrite [tex]\( g(x) \)[/tex] as:
[tex]\[
g(x) = (x - 6)(x + 6)(x + 7)
\][/tex]
3. Set the function equal to zero:
To find the zeros, we need to solve:
[tex]\[
(x - 6)(x + 6)(x + 7) = 0
\][/tex]
4. Find the roots:
The equation will be zero when any of the factors are zero:
[tex]\[
x - 6 = 0 \implies x = 6
\][/tex]
[tex]\[
x + 6 = 0 \implies x = -6
\][/tex]
[tex]\[
x + 7 = 0 \implies x = -7
\][/tex]
5. List the zeros of the function:
The zeros of the function are [tex]\( x = 6 \)[/tex], [tex]\( x = -6 \)[/tex], and [tex]\( x = -7 \)[/tex].
6. Check the given options:
The options are:
- 0
- -6
- 18
- 7
Among the options, the number [tex]\( -6 \)[/tex] is one of the zeros of the function.
Thus, the correct answer is:
[tex]\[
\boxed{-6}
\][/tex]
