On dividing `(x^(3)-6x+7)` by `(x+1)`, then the remainder is :

Options

A. 2
B. 0
C. 12
D. 6

Answer

To find the remainder when dividing the polynomial \( x^3 - 6x + 7 \) by \( x + 1 \), we can use polynomial long division. Here are the steps to solve the problem:

### Step-by-Step Solution:

1. **Set up the division**: We are dividing \( x^3 - 6x + 7 \) by \( x + 1 \). Since there is no \( x^2 \) term in the polynomial, we can rewrite it as \( x^3 + 0x^2 - 6x + 7 \).

2. **Divide the leading term**: Divide the leading term of the dividend \( x^3 \) by the leading term of the divisor \( x \):
\[ \frac{x^3}{x} = x^2 \]

3. **Multiply and subtract**: Multiply \( x^2 \) by the entire divisor \( x + 1 \):
\[ x^2(x + 1) = x^3 + x^2 \]
Now, subtract this from the original polynomial:
\[ (x^3 + 0x^2 - 6x + 7) - (x^3 + x^2) = -x^2 - 6x + 7 \]

4. **Repeat the process**: Now, divide the leading term of the new polynomial \( -x^2 \) by the leading term of the divisor \( x \):
\[ \frac{-x^2}{x} = -x \]
Multiply \( -x \) by \( x + 1 \):
\[ -x(x + 1) = -x^2 - x \]
Subtract this from the current polynomial:
\[ (-x^2 - 6x + 7) - (-x^2 - x) = -5x + 7 \]

5. **Continue the division**: Divide the leading term \( -5x \) by the leading term \( x \):
\[ \frac{-5x}{x} = -5 \]
Multiply \( -5 \) by \( x + 1 \):
\[ -5(x + 1) = -5x - 5 \]
Subtract this from the current polynomial:
\[ (-5x + 7) - (-5x - 5) = 12 \]

6. **Conclusion**: Since the degree of the remainder \( 12 \) is less than the degree of the divisor \( x + 1 \), we stop here. The remainder when \( x^3 - 6x + 7 \) is divided by \( x + 1 \) is:
\[ \text{Remainder} = 12 \]

### Final Answer:
The remainder is \( 12 \).