IDNLearn.com connects you with a community of experts ready to answer your questions. Find the answers you need quickly and accurately with help from our knowledgeable and dedicated community members.

What are the zeros of the function below? Check all that apply.

[tex]\[ F(x) = \frac{x(x+7)(x+2)}{(x-4)(x+4)} \][/tex]

A. 7
B. -7
C. -4
D. 0
E. -2
F. 4


Sagot :

To find the zeros of the function [tex]\( F(x) = \frac{x(x+7)(x+2)}{(x-4)(x+4)} \)[/tex], we need to determine the values of [tex]\( x \)[/tex] that make [tex]\( F(x) = 0 \)[/tex].

1. Identify the numerator and the denominator:

The numerator of [tex]\( F(x) \)[/tex] is [tex]\( x(x+7)(x+2) \)[/tex].
The denominator of [tex]\( F(x) \)[/tex] is [tex]\( (x-4)(x+4) \)[/tex].

2. Find the zeros of the numerator:

The function [tex]\(\frac{P(x)}{Q(x)} = 0\)[/tex] when [tex]\(P(x) = 0\)[/tex].

For [tex]\(x(x+7)(x+2) = 0\)[/tex]:
[tex]\[ x = 0, \quad x + 7 = 0 \implies x = -7, \quad x + 2 = 0 \implies x = -2 \][/tex]
Therefore, the potential zeros of the function are [tex]\( x = 0, x = -7, x = -2 \)[/tex].

3. Ensure these zeros do not make the denominator zero:

Evaluate the denominator at the potential zeros:
[tex]\[ \text{Denominator} = (x-4)(x+4) \][/tex]
- For [tex]\( x = 0 \)[/tex], the denominator is [tex]\( (0-4)(0+4) = (-4)(4) = -16 \neq 0 \)[/tex], so [tex]\( x = 0 \)[/tex] is valid.
- For [tex]\( x = -7 \)[/tex], the denominator is [tex]\( (-7-4)(-7+4) = (-11)(-3) = 33 \neq 0 \)[/tex], so [tex]\( x = -7 \)[/tex] is valid.
- For [tex]\( x = -2 \)[/tex], the denominator is [tex]\( (-2-4)(-2+4) = (-6)(2) = -12 \neq 0 \)[/tex], so [tex]\( x = -2 \)[/tex] is valid.

4. Exclude values that make the denominator zero:

The values that make the denominator zero are not included as zeros of the whole function.
Solve for:
[tex]\[ (x-4)(x+4) = 0 \implies x-4 = 0 \implies x = 4 \text{ and } x+4 = 0 \implies x = -4 \][/tex]
We exclude [tex]\( x = -4 \)[/tex] and [tex]\( x = 4 \)[/tex] from the potential zeros.

5. List the valid zeros:

After excluding the invalid denominations, the valid zeros of the function are:
[tex]\[ x = 0, x = -7, x = -2 \][/tex]

So, checking the provided options:
- B. [tex]\( -7 \)[/tex]
- D. [tex]\( 0 \)[/tex]
- E. [tex]\( -2 \)[/tex]

These are the correct zeros of the function.
Thank you for participating in our discussion. We value every contribution. Keep sharing knowledge and helping others find the answers they need. Let's create a dynamic and informative learning environment together. Thank you for choosing IDNLearn.com. We’re committed to providing accurate answers, so visit us again soon.