Select the correct answer.

Which statement best describes the zeros of the function [tex]h(x)=(x+9)\left(x^2-10x+25\right)[/tex]?

A. The function has three complex zeros.
B. The function has three distinct real zeros.
C. The function has one real zero and two complex zeros.
D. The function has two distinct real zeros.



Answer :

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.