First, let's analyze the function [tex]\( h(x) = (x + 9)(x^2 - 10x + 25) \)[/tex] to identify its zeros.
### Step-by-Step Solution:
1. Find the zeros of each factor:
- The first factor is [tex]\( x + 9 \)[/tex]. To find its zero:
[tex]\[
x + 9 = 0 \implies x = -9
\][/tex]
So, one zero is [tex]\( x = -9 \)[/tex].
- The second factor is [tex]\( x^2 - 10x + 25 \)[/tex]. This can be solved by recognizing that it's a perfect square trinomial:
[tex]\[
x^2 - 10x + 25 = (x - 5)^2
\][/tex]
To find the zero:
[tex]\[
(x - 5)^2 = 0 \implies x - 5 = 0 \implies x = 5
\][/tex]
So, the zero for this factor is [tex]\( x = 5 \)[/tex].
2. Analyze the zeros:
- The zero [tex]\( x = -9 \)[/tex] is real.
- The zero [tex]\( x = 5 \)[/tex] is real, but since it's from [tex]\((x - 5)^2\)[/tex], it’s a repeated root, meaning it has a multiplicity of 2.
3. Count the distinct real zeros:
- There are two distinct real zeros: [tex]\( x = -9 \)[/tex] and [tex]\( x = 5 \)[/tex].
4. Check for complex zeros:
- Since both zeros we found are real and there are no complex solutions in the factorization, there are no complex zeros.
### Conclusion:
- The function [tex]\( h(x) = (x + 9)(x^2 - 10x + 25) \)[/tex] has two distinct real zeros.
Therefore, the correct answer is:
D. The function has two distinct real zeros.
