When the polynomial expression 2x^3-3x^2-cx+d is divided by (x+1) and (x-2), the same remainder of 64 is obtained. Determine the value of c and d. Show working step by step using the remainder factor theorem.

To solve for \( c \) and \( d \) in the polynomial \( f(x) = 2x^3 - 3x^2 - cx + d \) given that dividing it by \( (x+1) \) and \( (x-2) \) leaves the same remainder of 64, we can use the **Remainder Theorem**.

### Step 1: Apply the Remainder Theorem
According to the Remainder Theorem, if a polynomial \( f(x) \) is divided by \( (x - a) \), the remainder of the division is \( f(a) \).

### Step 2: Set Up the Equations
1. When \( f(x) \) is divided by \( (x+1) \), which is \( x - (-1) \), we substitute \( x = -1 \):
   \[
   f(-1) = 2(-1)^3 - 3(-1)^2 - c(-1) + d = 64
   \]
   Simplifying:
   \[
   f(-1) = 2(-1) - 3(1) + c + d = 64
   \]
   \[
   -2 - 3 + c + d = 64
   \]
   \[
   c + d = 64 + 5 = 69
   \]
   So, we have our first equation:
   \[
   c + d = 69
   \]

2. When \( f(x) \) is divided by \( (x - 2) \), we substitute \( x = 2 \):
   \[
   f(2) = 2(2)^3 - 3(2)^2 - c(2) + d = 64
   \]
   Simplifying:
   \[
   f(2) = 2(8) - 3(4) - 2c + d = 64
   \]
   \[
   16 - 12 - 2c + d = 64
   \]
   \[
   -2c + d = 64 - 4 = 68
   \]
   This gives our second equation:
   \[
   -2c + d = 68
   \]

### Step 3: Solve the System of Equations
Now we have two equations:
1. \( c + d = 69 \)
2. \( -2c + d = 68 \)

Let's solve for \( c \) and \( d \) by eliminating \( d \).

1. Subtract the first equation from the second equation to eliminate \( d \):
   \[
   (-2c + d) - (c + d) = 68 - 69
   \]
   \[
   -3c = -1
   \]
   \[
   c = \frac{-1}{-3} = \frac{1}{3}
   \]

2. Substitute \( c = \frac{1}{3} \) into the first equation to solve for \( d \):
   \[
   \frac{1}{3} + d = 69
   \]
   \[
   d = 69 - \frac{1}{3} = \frac{207 - 1}{3} = \frac{206}{3}
   \]

### Final Answer
The values of \( c \) and \( d \) are:
\[
c = \frac{1}{3}, \quad d = \frac{206}{3}
\]

Would you like further details on any of these steps?

Here are five related questions for further exploration:
1. What is the Remainder Theorem, and how does it apply to polynomial division?
2. How would the solution change if different remainders were given for \( x+1 \) and \( x-2 \)?
3. Can we generalize this solution to find unknown coefficients for higher-degree polynomials?
4. How does polynomial long division relate to the Remainder Theorem?
5. What other methods can be used to determine the coefficients in polynomials?

**Tip:** When working with equations derived from polynomial roots or remainders, ensure all signs are correctly managed, as a minor sign error can lead to significant mistakes in the final answer.

Explore a step-by-step solution to finding the values of c and d in the polynomial 2x^3 - 3x^2 - cx + d. Using the Remainder Theorem, this problem shows how to set up and solve algebraic equations for polynomial remainders. Ideal for high school students studying polynomials.

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