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The image contains two questions asking to evaluate a polynomial using the **Remainder Theorem**. Let's solve them step-by-step.

### Question 11:
Use the Remainder Theorem to evaluate \( P(c) \).

Given:
\[ P(x) = x^3 - 9x^2 + 14 \]
and \( c = 2 \).

The **Remainder Theorem** states that if a polynomial \( P(x) \) is divided by \( x - c \), then the remainder is \( P(c) \). Therefore, we can simply substitute \( c = 2 \) into \( P(x) \):

\[
P(2) = (2)^3 - 9(2)^2 + 14
\]
\[
P(2) = 8 - 9(4) + 14
\]
\[
P(2) = 8 - 36 + 14
\]
\[
P(2) = -14
\]

So, the answer for Question 11 is:
\[
P(2) = -14
\]

---

### Question 12:
Use the Remainder Theorem to evaluate \( f(c) \).

Given:
\[ P(x) = x^4 + 9x^3 - 8x - 15 \]
and \( c = -1 \).

Substitute \( c = -1 \) into \( P(x) \):

\[
P(-1) = (-1)^4 + 9(-1)^3 - 8(-1) - 15
\]
\[
P(-1) = 1 + 9(-1) + 8 - 15
\]
\[
P(-1) = 1 - 9 + 8 - 15
\]
\[
P(-1) = -15
\]

So, the answer for Question 12 is:
\[
P(-1) = -15
\]

---

Let me know if you'd like more details or explanations!

### Related Questions:
1. How is the Remainder Theorem related to the Factor Theorem?
2. What is the significance of the degree of a polynomial when using the Remainder Theorem?
3. Can you apply synthetic division to verify the result of the Remainder Theorem?
4. How do we handle polynomials with complex coefficients using the Remainder Theorem?
5. How can we interpret the Remainder Theorem geometrically on a graph?

### Tip:
Always double-check your substitution and arithmetic when applying the Remainder Theorem to ensure accuracy.

Use the Remainder Theorem to evaluate P(c). Given: P(x) = x^3 - 9x^2 + 14, c = 2. Use the Remainder Theorem to evaluate f(c). Given: P(x) = x^4 + 9x^3 - 8x - 15, c = -1.

Solve polynomial problems using the Remainder Theorem. This example walks through evaluating P(2) for P(x) = x^3 - 9x^2 + 14 and P(-1) for P(x) = x^4 + 9x^3 - 8x - 15. A detailed solution is provided with step-by-step explanations.

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