If `x^(4) - 8x^(3) + x^(2) + 3x-6` is divided by (x+2), then find the remainder.

Options

A. 45
B. 50
C. 48
D. 72

Answer

To find the remainder when the polynomial \( f(x) = x^4 - 8x^3 + x^2 + 3x - 6 \) is divided by \( x + 2 \), we can use the Remainder Theorem. According to this theorem, the remainder of the division of a polynomial \( f(x) \) by \( x - c \) is \( f(c) \). In our case, we need to evaluate \( f(-2) \) since we are dividing by \( x + 2 \).

### Step-by-Step Solution:

1. **Identify the Polynomial and the Value to Substitute:**
We have the polynomial:
\[ f(x) = x^4 - 8x^3 + x^2 + 3x - 6 \]
We need to find \( f(-2) \).

2. **Substitute \( -2 \) into the Polynomial:**
\[ f(-2) = (-2)^4 - 8(-2)^3 + (-2)^2 + 3(-2) - 6 \]

3. **Calculate Each Term:**
- Calculate \( (-2)^4 \):
\[ (-2)^4 = 16 \]
- Calculate \( -8(-2)^3 \):
\[ (-2)^3 = -8 \quad \Rightarrow \quad -8(-8) = 64 \]
- Calculate \( (-2)^2 \):
\[ (-2)^2 = 4 \]
- Calculate \( 3(-2) \):
\[ 3(-2) = -6 \]
- The constant term is \( -6 \).

4. **Combine All the Terms:**
Now substitute these values back into the equation:
\[ f(-2) = 16 + 64 + 4 - 6 - 6 \]
Combine the terms:
\[ f(-2) = 16 + 64 + 4 - 6 - 6 = 16 + 64 + 4 - 12 \]
\[ f(-2) = 80 - 12 = 68 \]

5. **Final Result:**
The remainder when \( f(x) \) is divided by \( x + 2 \) is:
\[ \text{Remainder} = 68 \]