To solve the problem, we need to find the zeros of the function [tex]\( h(x) = (x - 6)(x^2 + 8x + 16) \)[/tex] and determine their nature.
1. Factor the quadratic expression [tex]\( x^2 + 8x + 16 \)[/tex]:
The quadratic expression can be factored as follows:
[tex]\[
x^2 + 8x + 16 = (x + 4)^2
\][/tex]
Therefore, the function [tex]\( h(x) \)[/tex] can be rewritten as:
[tex]\[
h(x) = (x - 6)(x + 4)^2
\][/tex]
2. Find the zeros of [tex]\( h(x) \)[/tex]:
The zeros of the function occur when each factor is equal to zero.
[tex]\[
x - 6 = 0 \quad \Rightarrow \quad x = 6
\][/tex]
[tex]\[
(x + 4)^2 = 0 \quad \Rightarrow \quad x + 4 = 0 \quad \Rightarrow \quad x = -4
\][/tex]
Here, [tex]\( x = -4 \)[/tex] is a repeated zero (it has multiplicity 2).
3. Determine the nature and distinctness of the zeros:
The zeros of [tex]\( h(x) \)[/tex] are:
[tex]\[
x = 6 \quad \text{and} \quad x = -4 \, (repeated)
\][/tex]
So, the zeros are [tex]\( x = 6 \)[/tex] and [tex]\( x = -4 \)[/tex]. Even though [tex]\( x = -4 \)[/tex] is repeated, it is still considered as one distinct zero. Hence, there are two distinct real zeros: [tex]\( x = 6 \)[/tex] and [tex]\( x = -4 \)[/tex].
4. Select the correct statement:
The function [tex]\( h(x) \)[/tex] has exactly two distinct real zeros: [tex]\( x = 6 \)[/tex] and [tex]\( x = -4 \)[/tex] (with multiplicity 2).
The correct answer is:
A. The function has two distinct real zeros.
