If $6x^3 + 2x^2 - 5x + 1$ divides $x^2 - x - 1$, find the remainder.

To solve this problem, we need to perform polynomial long division. We are dividing the polynomial $P(x) = 6x^3 + 2x^2 - 5x + 1$ by the divisor $D(x) = x^2 - x - 1$.

1. Analyze the question

We are dividing a cubic polynomial by a quadratic polynomial. The result will be a linear quotient and a linear remainder (since the degree of the remainder must be less than the degree of the divisor).

2. Explain the concept

Polynomial long division follows the same logic as numerical long division:

• Divide the leading term of the dividend by the leading term of the divisor.
• Multiply the entire divisor by the result.
• Subtract the result from the current dividend.
• Repeat until the degree of the remainder is less than the degree of the divisor.

3. Show the solution

Step 1: Divide the first terms Divide $6x^3$ by $x^2$: $6x^3 / x^2 = 6x$

Step 2: Multiply and Subtract Multiply $6x$ by $(x^2 - x - 1)$: $6x(x^2 - x - 1) = 6x^3 - 6x^2 - 6x$

Subtract this from the original polynomial: $(6x^3 + 2x^2 - 5x + 1) - (6x^3 - 6x^2 - 6x)$ $= (6x^3 - 6x^3) + (2x^2 - (-6x^2)) + (-5x - (-6x)) + 1$ $= 8x^2 + x + 1$

Step 3: Divide the new leading term Divide $8x^2$ by $x^2$: $8x^2 / x^2 = 8$

Step 4: Multiply and Subtract Multiply $8$ by $(x^2 - x - 1)$: $8(x^2 - x - 1) = 8x^2 - 8x - 8$

Subtract this from the current remainder ($8x^2 + x + 1$): $(8x^2 + x + 1) - (8x^2 - 8x - 8)$ $= (8x^2 - 8x^2) + (x - (-8x)) + (1 - (-8))$ $= 9x + 9$

4. Identify the answer

The remainder is $9x + 9$. This matches option A.

5. Exam Tip

When performing polynomial subtraction, be very careful with the signs, especially when subtracting a negative term (e.g., $-(-6x^2)$ becomes $+6x^2$). A common mistake in JAMB exams is losing a negative sign during this step. Always write out the subtraction clearly to avoid "silly" errors.