Find the sum of remainders when `x^(3)-3x^(2)+4x-4` is divided by (x-1) and (x+2).

Answer

To solve the problem of finding the sum of the remainders when the polynomial \( p(x) = x^3 - 3x^2 + 4x - 4 \) is divided by \( (x-1) \) and \( (x+2) \), we can follow these steps:

### Step 1: Find the remainder when \( p(x) \) is divided by \( (x-1) \)
To find the remainder when dividing by \( (x-1) \), we evaluate \( p(1) \).

\[ p(1) = 1^3 - 3(1^2) + 4(1) - 4 \]
\[ = 1 - 3 + 4 - 4 \]
\[ = 1 - 3 + 4 - 4 = -2 \]

### Step 2: Find the remainder when \( p(x) \) is divided by \( (x+2) \)
To find the remainder when dividing by \( (x+2) \), we evaluate \( p(-2) \).

\[ p(-2) = (-2)^3 - 3(-2)^2 + 4(-2) - 4 \]
\[ = -8 - 3(4) - 8 - 4 \]
\[ = -8 - 12 - 8 - 4 \]
\[ = -32 \]

### Step 3: Calculate the sum of the remainders
Now, we add the two remainders we found:

\[ \text{Sum of remainders} = p(1) + p(-2) \]
\[ = -2 + (-32) \]
\[ = -2 - 32 = -34 \]

### Final Answer
The sum of the remainders when \( p(x) \) is divided by \( (x-1) \) and \( (x+2) \) is \( -34 \).

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