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The problem in the image is asking to find the indicated value of the polynomial \( P(x) = 2x^5 + 3x^2 - 2x + 1 \) at \( x = -2 \), using the Remainder Theorem.

The Remainder Theorem states that if a polynomial \( P(x) \) is divided by \( (x - a) \), the remainder of that division is \( P(a) \). In this case, we are asked to find \( P(-2) \).

Let's evaluate \( P(x) \) at \( x = -2 \):

\[
P(x) = 2x^5 + 3x^2 - 2x + 1
\]

Substitute \( x = -2 \):

\[
P(-2) = 2(-2)^5 + 3(-2)^2 - 2(-2) + 1
\]

Simplify step-by-step:

\[
P(-2) = 2(-32) + 3(4) - 2(-2) + 1
\]

\[
P(-2) = -64 + 12 + 4 + 1
\]

\[
P(-2) = -64 + 17
\]

\[
P(-2) = -47
\]

Thus, the value of the polynomial \( P(x) \) at \( x = -2 \) is \( -47 \).

Would you like more details or have any questions?

Here are some related questions you might explore:
1. How can you apply the Remainder Theorem to different polynomials?
2. What is the Factor Theorem, and how does it relate to the Remainder Theorem?
3. How would synthetic division be used to verify this result?
4. What is the significance of evaluating polynomials at specific points?
5. How can graphing polynomials help visualize the result of the Remainder Theorem?

**Tip:** When using the Remainder Theorem, always check if the polynomial can be simplified further before substituting the value for quick computation.

Find the indicated value of the polynomial P(x) = 2x^5 + 3x^2 - 2x + 1 using the Remainder Theorem; find P(-2).

Learn how to apply the Remainder Theorem to evaluate the polynomial P(x) = 2x^5 + 3x^2 - 2x + 1 at x = -2. This step-by-step guide demonstrates key algebraic concepts and simplification techniques, suitable for high school students in Grades 10-12.

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