For what value of c will the polynomial
P(x) = -2x3 + cx2 - 5x + 2 have the same
remainder when it is divided by x - 2 and
by x + 1?

We are given the polynomial \( P(x) = -2x^3 + cx^2 - 5x + 2 \), and we need to find the value of \( c \) such that the remainder is the same when divided by \( x - 2 \) and \( x + 1 \).

According to the **Remainder Theorem**, the remainder when a polynomial \( P(x) \) is divided by \( x - a \) is \( P(a) \). Therefore, the remainder when divided by \( x - 2 \) is \( P(2) \), and the remainder when divided by \( x + 1 \) is \( P(-1) \).

We are told that these remainders are equal, so:

\[
P(2) = P(-1)
\]

### Step 1: Calculate \( P(2) \)
Substitute \( x = 2 \) into \( P(x) \):

\[
P(2) = -2(2)^3 + c(2)^2 - 5(2) + 2
\]
\[
P(2) = -2(8) + c(4) - 10 + 2
\]
\[
P(2) = -16 + 4c - 10 + 2
\]
\[
P(2) = 4c - 24
\]

### Step 2: Calculate \( P(-1) \)
Substitute \( x = -1 \) into \( P(x) \):

\[
P(-1) = -2(-1)^3 + c(-1)^2 - 5(-1) + 2
\]
\[
P(-1) = -2(-1) + c(1) + 5 + 2
\]
\[
P(-1) = 2 + c + 5 + 2
\]
\[
P(-1) = c + 9
\]

### Step 3: Set \( P(2) = P(-1) \)
Now, set the two expressions equal to each other:

\[
4c - 24 = c + 9
\]

### Step 4: Solve for \( c \)
Subtract \( c \) from both sides:

\[
3c - 24 = 9
\]

Add 24 to both sides:

\[
3c = 33
\]

Finally, divide by 3:

\[
c = 11
\]

Thus, the value of \( c \) that makes the remainders equal is \( \boxed{11} \).

Would you like further details or have any questions?

Here are 5 related questions to explore:
1. How does the Remainder Theorem work for polynomials?
2. What is the significance of the factorization of a polynomial?
3. How would the remainder change if divided by other linear factors like \(x - 3\)?
4. Can synthetic division be used to solve similar problems, and how?
5. How does the value of \( c \) affect the graph of the polynomial?

**Tip:** When working with polynomials and the Remainder Theorem, always substitute carefully to avoid errors, especially with negative signs.

For what value of c will the polynomial P(x) = -2x^3 + cx^2 - 5x + 2 have the same remainder when it is divided by x - 2 and by x + 1?

Learn how to find the value of c in the polynomial P(x) = -2x^3 + cx^2 - 5x + 2, where the remainder is the same when divided by x - 2 and x + 1. This step-by-step solution involves the Remainder Theorem and algebraic substitution, suitable for students in Grades 10-12.

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