On dividing `(x^3-5x + 6)` by(x+1) the the remainder is:

Options

A. 2
B. 12
C. 10
D. 6

Answer

To find the remainder when dividing \( x^3 - 5x + 6 \) by \( x + 1 \), we can use polynomial long division. Here’s a step-by-step solution:

### Step 1: Set up the division
We want to divide \( x^3 - 5x + 6 \) by \( x + 1 \).

### Step 2: Divide the leading term
Divide the leading term of the dividend \( x^3 \) by the leading term of the divisor \( x \):
\[ \frac{x^3}{x} = x^2 \]
So, the first term of our quotient is \( x^2 \).

### Step 3: Multiply and subtract
Now, multiply \( x^2 \) by the entire divisor \( x + 1 \):
\[ x^2(x + 1) = x^3 + x^2 \]
Subtract this from the original polynomial:
\[ (x^3 - 5x + 6) - (x^3 + x^2) = -x^2 - 5x + 6 \]

### Step 4: Repeat the process
Now, take the new polynomial \( -x^2 - 5x + 6 \) and repeat the process. Divide the leading term \( -x^2 \) by \( x \):
\[ \frac{-x^2}{x} = -x \]
So, the next term of our quotient is \( -x \).

### Step 5: Multiply and subtract again
Multiply \( -x \) by the divisor \( x + 1 \):
\[ -x(x + 1) = -x^2 - x \]
Subtract this from the current polynomial:
\[ (-x^2 - 5x + 6) - (-x^2 - x) = -4x + 6 \]

### Step 6: Repeat again
Now, take \( -4x + 6 \) and divide \( -4x \) by \( x \):
\[ \frac{-4x}{x} = -4 \]
So, the next term of our quotient is \( -4 \).

### Step 7: Multiply and subtract once more
Multiply \( -4 \) by the divisor \( x + 1 \):
\[ -4(x + 1) = -4x - 4 \]
Subtract this from the current polynomial:
\[ (-4x + 6) - (-4x - 4) = 10 \]

### Conclusion
The remainder when dividing \( x^3 - 5x + 6 \) by \( x + 1 \) is \( 10 \).

### Final Answer
The remainder is \( 10 \).

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