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 \( x^4 - 8x^3 + x^2 + 3x - 6 \) is divided by \( x + 2 \), we can use polynomial long division. Here’s a step-by-step solution:

### Step 1: Set up the division
We want to divide \( x^4 - 8x^3 + x^2 + 3x - 6 \) by \( x + 2 \).

### Step 2: Divide the leading terms
Divide the leading term of the dividend \( x^4 \) by the leading term of the divisor \( x \):
\[ \frac{x^4}{x} = x^3 \]
So, we write \( x^3 \) above the division line.

### Step 3: Multiply and subtract
Now, multiply \( x^3 \) by \( x + 2 \):
\[ x^3(x + 2) = x^4 + 2x^3 \]
Subtract this from the original polynomial:
\[ (x^4 - 8x^3 + x^2 + 3x - 6) - (x^4 + 2x^3) = -10x^3 + x^2 + 3x - 6 \]

### Step 4: Repeat the process
Now, divide the new leading term \( -10x^3 \) by \( x \):
\[ \frac{-10x^3}{x} = -10x^2 \]
Write \( -10x^2 \) above the division line.

### Step 5: Multiply and subtract again
Multiply \( -10x^2 \) by \( x + 2 \):
\[ -10x^2(x + 2) = -10x^3 - 20x^2 \]
Subtract this from the current polynomial:
\[ (-10x^3 + x^2 + 3x - 6) - (-10x^3 - 20x^2) = 21x^2 + 3x - 6 \]

### Step 6: Continue the process
Now, divide \( 21x^2 \) by \( x \):
\[ \frac{21x^2}{x} = 21x \]
Write \( 21x \) above the division line.

### Step 7: Multiply and subtract again
Multiply \( 21x \) by \( x + 2 \):
\[ 21x(x + 2) = 21x^2 + 42x \]
Subtract this from the current polynomial:
\[ (21x^2 + 3x - 6) - (21x^2 + 42x) = -39x - 6 \]

### Step 8: Final division
Now, divide \( -39x \) by \( x \):
\[ \frac{-39x}{x} = -39 \]
Write \( -39 \) above the division line.

### Step 9: Multiply and subtract one last time
Multiply \( -39 \) by \( x + 2 \):
\[ -39(x + 2) = -39x - 78 \]
Subtract this from the current polynomial:
\[ (-39x - 6) - (-39x - 78) = 72 \]

### Conclusion
The remainder when \( x^4 - 8x^3 + x^2 + 3x - 6 \) is divided by \( x + 2 \) is \( 72 \).

### Final Answer
The remainder is \( 72 \).