To solve this problem, let's analyze the given function step by step.

Given Function:

$p(x) = \frac{x^2 + 10x - 11}{x - 11}$

We are tasked to determine for the given values of $x$ whether $p(x)$ has a zero, a vertical asymptote, or a removable discontinuity.

Step 1: Factorize the numerator

The numerator is $x^2 + 10x - 11$. Factoring it: $x^2 + 10x - 11 = (x + 11)(x - 1)$

So the function becomes: $p(x) = \frac{(x + 11)(x - 1)}{x - 11}$

Step 2: Analyze the denominator and numerator

• The denominator $x - 11$ causes the function to be undefined at $x = 11$.
• The roots of the numerator, $x + 11 = 0$ and $x - 1 = 0$, are $x = -11$ and $x = 1$, respectively. These are potential zeros of the function (if they don't cancel out with the denominator).

Step 3: Check the given values of $x$

(a) $x = -11$

• Substituting $x = -11$ into the denominator: $x - 11 = -11 - 11 = -22 \neq 0$
• Since the denominator is non-zero, and the numerator becomes zero ($(-11 + 11)(-11 - 1) = 0$), this value is a zero of the function.

(b) $x = 1$

• Substituting $x = 1$ into the denominator: $x - 11 = 1 - 11 = -10 \neq 0$
• Since the denominator is non-zero, and the numerator becomes zero ($(1 + 11)(1 - 1) = 0$), this value is also a zero of the function.

(c) $x = 11$

• Substituting $x = 11$ into the denominator: $x - 11 = 11 - 11 = 0$
• The function is undefined at $x = 11$. Looking at the factored form of $p(x)$, the term $x - 11$ does not cancel with any numerator term. Hence, $x = 11$ causes a vertical asymptote.

Final Answer:

• $x = -11$: Zero
• $x = 1$: Zero
• $x = 11$: Vertical Asymptote

Would you like a detailed explanation of any part? Here are some follow-up questions:

1. How can you determine zeros of a rational function?
2. What is the difference between a removable discontinuity and a vertical asymptote?
3. How do you factorize quadratic expressions quickly?
4. Why does $x = 11$ not create a removable discontinuity?
5. Can there be multiple vertical asymptotes in a rational function?

Tip: Always factorize the numerator and denominator to simplify rational functions before analyzing discontinuities or zeros!