To determine the zeros of the quadratic function [tex]\( g \)[/tex] given its factors [tex]\( (x - 7) \)[/tex] and [tex]\( (x + 3) \)[/tex], follow these steps:
1. Identify the factors: The function [tex]\( g(x) \)[/tex] is factored as [tex]\( (x - 7) \)[/tex] and [tex]\( (x + 3) \)[/tex]. This means the function can be written as:
[tex]\[
g(x) = (x - 7)(x + 3)
\][/tex]
2. Set each factor equal to zero: The zeros of the function are the values of [tex]\( x \)[/tex] that make each factor equal to zero.
- For the factor [tex]\( (x - 7) \)[/tex], set it equal to zero:
[tex]\[
x - 7 = 0
\][/tex]
Solving for [tex]\( x \)[/tex], we get:
[tex]\[
x = 7
\][/tex]
- For the factor [tex]\( (x + 3) \)[/tex], set it equal to zero:
[tex]\[
x + 3 = 0
\][/tex]
Solving for [tex]\( x \)[/tex], we get:
[tex]\[
x = -3
\][/tex]
3. Combine the zeros: The zeros of the function [tex]\( g(x) \)[/tex] are the solutions to the above equations. Therefore, the zeros are:
[tex]\[
x = 7 \quad \text{and} \quad x = -3
\][/tex]
Thus, the correct answer is:
D. [tex]\(-3\)[/tex] and [tex]\(7\)[/tex]
