Here is the breakdown of the provided math problems and explanations for the questions visible in the image:

Question 6:

The zeros of $g(x) = x^2 - 1$ are the _________ of $f(x) = \frac{x^2 - 1}{(x-1)^2}$.

1. First, factorize $x^2 - 1$:
   $x^2 - 1 = (x - 1)(x + 1)$
   So, $g(x)$ has zeros at $x = 1$ and $x = -1$.

2. Now analyze $f(x)$:
   $f(x) = \frac{x^2 - 1}{(x-1)^2} = \frac{(x-1)(x+1)}{(x-1)^2}$

   Simplify $f(x)$:
   f(x) = \frac{x+1}{x-1}, \quad \text{(as long as x \neq 1)}.

3. At $x = 1$:

   • The denominator becomes zero, but the numerator does not eliminate the $(x-1)$ term entirely.
   • Therefore, there is a hole at $x = 1$.

4. At $x = -1$:

   • The numerator equals zero, and the denominator is not zero.
   • Therefore, there is an $x$-intercept at $x = -1$.

Correct Answer:

• B. hole at $x = 1$.
• A. x-intercepts at $x = -1$.

Question 7:

When a polynomial $P(x)$ is divided by $4x - 1$, the remainder is $-10$.

This is based on the Remainder Theorem:

• If $P(x)$ is divided by $ax - b$, then the remainder is $P\left(\frac{b}{a}\right)$.
• Here $ax - b = 4x - 1 \Rightarrow x = \frac{1}{4}$.

Thus, the remainder is given as:
$P\left(\frac{1}{4}\right) = -10.$

Correct Answer:

• D. -10

Question 8:

What is the number of turning points in the polynomial $f(x) = 5(x - a)(x - b)(x - c)^2$, where $c > b > a$?

1. The number of turning points is determined by the degree of the polynomial:

   • $(x - a)$ and $(x - b)$ are linear factors (degree $1$).
   • $(x - c)^2$ is a quadratic factor (degree $2$).
   • Therefore, the degree of $f(x)$ is $1 + 1 + 2 = 4$.

2. The maximum number of turning points in a polynomial of degree $n$ is $n - 1$:

   • Here $n = 4$, so the maximum number of turning points is $4 - 1 = 3$.

Correct Answer:

• A. 3

Recap of Answers:

1. 6: B. hole
2. 7: D. -10
3. 8: A. 3

Follow-Up Questions:

1. How do you determine holes and asymptotes in rational functions?
2. What is the Remainder Theorem, and how is it applied?
3. Why does a polynomial of degree $n$ have at most $n-1$ turning points?
4. Can a polynomial have fewer turning points than the maximum possible? Why?
5. How do you factorize and analyze rational functions like $f(x)$?

Tip:

To identify holes in a rational function, look for factors in both the numerator and denominator that cancel out—this is where the function becomes undefined (a "hole").