To determine the zeros of the function [tex]\( F(x) = \frac{(x + 3)(x - 1)}{(x - 2)(x + 2)} \)[/tex], we need to find the values of [tex]\( x \)[/tex] that make [tex]\( F(x) = 0 \)[/tex]. For a rational function, the zeros occur where the numerator is zero, provided the denominator is not zero at those points.
Here are the steps to find the zeros of the function:
1. Identify the numerator:
The numerator of the function is [tex]\( (x + 3)(x - 1) \)[/tex].
2. Set the numerator equal to zero:
To find the zeros, we solve the equation:
[tex]\[
(x + 3)(x - 1) = 0
\][/tex]
3. Solve for [tex]\( x \)[/tex]:
The solutions to the equation [tex]\( (x + 3)(x - 1) = 0 \)[/tex] are found by setting each factor equal to zero:
[tex]\[
x + 3 = 0 \quad \text{or} \quad x - 1 = 0
\][/tex]
Solving these, we get:
[tex]\[
x = -3 \quad \text{and} \quad x = 1
\][/tex]
4. Check the denominator:
We must ensure that these potential zeros do not make the denominator [tex]\( (x - 2)(x + 2) \)[/tex] equal to zero, as this would make the function undefined.
[tex]\[
x - 2 = 0 \quad \Rightarrow \quad x = 2
\][/tex]
[tex]\[
x + 2 = 0 \quad \Rightarrow \quad x = -2
\][/tex]
The values [tex]\( x = 2 \)[/tex] and [tex]\( x = -2 \)[/tex] are excluded because they make the denominator zero, and thus these points cannot be zeros of the function.
After verifying, we confirm that [tex]\( x = -3 \)[/tex] and [tex]\( x = 1 \)[/tex] are indeed zeros of the function because they satisfy the condition that the numerator is zero and the denominator is not zero at these points.
Therefore, the correct zeros of the function [tex]\( F(x) = \frac{(x + 3)(x - 1)}{(x - 2)(x + 2)} \)[/tex] are:
- [tex]\( x = 1 \)[/tex]
- [tex]\( x = -3 \)[/tex]
Answer:
From the given options, the zeros of the function are:
- B. 1
- E. -3
