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.