To determine which equation could represent the function [tex]\( f \)[/tex] with zeros at [tex]\( -1 \)[/tex] and [tex]\( -5 \)[/tex], we need to understand that the zeros of a function are the values of [tex]\( x \)[/tex] that make the function equal to zero.
Given the zeros [tex]\( -1 \)[/tex] and [tex]\( -5 \)[/tex]:
1. The zero [tex]\( x = -1 \)[/tex] implies that one of the factors of the function is [tex]\( (x + 1) \)[/tex].
2. The zero [tex]\( x = -5 \)[/tex] implies that the other factor of the function is [tex]\( (x + 5) \)[/tex].
Thus, the function [tex]\( f(x) \)[/tex] can be written as the product of these factors:
[tex]\[ f(x) = (x + 1)(x + 5) \][/tex]
Let's confirm this form matches our given options:
1. [tex]\( f(x) = (x - 1)(x + 5) \)[/tex]
2. [tex]\( f(x) = (x + 1)(x - 5) \)[/tex]
3. [tex]\( f(x) = (x + 1)(x + 5) \)[/tex]
4. [tex]\( f(x) = (x - 1)(x - 5) \)[/tex]
Among these, the equation [tex]\( f(x) = (x + 1)(x + 5) \)[/tex] matches our required factors. Hence, the correct answer is:
[tex]\[ f(x) = (x + 1)(x + 5) \][/tex]
