To determine the zeros of the function [tex]\( F(x) = \frac{(x-1)(x+1)}{6(x-4)(x+7)} \)[/tex], we need to identify the values of [tex]\( x \)[/tex] at which the function equals zero.
A function [tex]\( F(x) \)[/tex] is zero when its numerator is zero, provided that the denominator is not also zero at the same points.
The numerator of [tex]\( F(x) \)[/tex] is:
[tex]\[
(x-1)(x+1)
\][/tex]
To find the zeros of this function, we need to solve the equation:
[tex]\[
(x-1)(x+1) = 0
\][/tex]
We can solve this by setting each factor equal to zero:
[tex]\[
x - 1 = 0 \quad \text{or} \quad x + 1 = 0
\][/tex]
Solving these equations separately:
[tex]\[
x - 1 = 0 \implies x = 1
\][/tex]
[tex]\[
x + 1 = 0 \implies x = -1
\][/tex]
Thus, the zeros of the function [tex]\( F(x) \)[/tex] are [tex]\( x = 1 \)[/tex] and [tex]\( x = -1 \)[/tex].
Therefore, checking the given answers:
- A. 4: This is not a zero.
- B. -1: This is a zero.
- C. 1: This is a zero.
- D. 6: This is not a zero.
- E. -7: This is not a zero.
- F. 7: This is not a zero.
So, the correct answers are:
B. -1 and C. 1
