To determine which expression could be a factor of the function [tex]\(d\)[/tex] given its zeros are -3 and 8, we need to find the corresponding factors associated with these zeros.
1. Understanding Zeros of a Function:
- A zero (or root) of a function [tex]\(d(x)\)[/tex] is a value [tex]\(a\)[/tex] such that [tex]\(d(a) = 0\)[/tex].
- If [tex]\(a\)[/tex] is a zero of the function, then [tex]\( (x - a)\)[/tex] is a factor of the function.
2. Identifying the Factors:
- For the zero at [tex]\(-3\)[/tex]:
- If [tex]\(-3\)[/tex] is a root, then the factor must be [tex]\((x - (-3)) = (x + 3)\)[/tex].
- For the zero at [tex]\(8\)[/tex]:
- If [tex]\(8\)[/tex] is a root, then the factor must be [tex]\((x - 8)\)[/tex].
3. Analyzing the Given Choices:
- A. [tex]\((x - 3)\)[/tex]:
- This factor would correspond to a zero at [tex]\(3\)[/tex], not [tex]\(-3\)[/tex]. Hence, this is not the correct factor.
- B. [tex]\((x + 8)\)[/tex]:
- This factor would correspond to a zero at [tex]\(-8\)[/tex], not [tex]\(8\)[/tex]. Hence, this is not the correct factor.
- C. [tex]\((x + 5)\)[/tex]:
- This factor would correspond to a zero at [tex]\(-5\)[/tex], which is irrelevant to the given zeros [tex]\(-3\)[/tex] and [tex]\(8\)[/tex]. Hence, this is not the correct factor.
- D. [tex]\((x + 3)\)[/tex]:
- This factor corresponds to a zero at [tex]\(-3\)[/tex]. Therefore, this is a correct factor.
Consequently, the correct expression that could be a factor of the function [tex]\(d(x)\)[/tex] is:
[tex]\[ D. (x + 3) \][/tex]
