Question

The remainder when the polynomial \[ 2x^5-3x^4+5x^3-3x^2+7x-9 \] is divided by \[ x^2-x-3 \] is

• A. \(-41x-3\)
• B. \(41x+3\)
• C. \(41x-3\)
• D. \(-41x+3\)

Hint:

When dividing by a quadratic polynomial, reduce higher powers using the relation obtained from the divisor. This avoids long division and quickly yields the remainder.

Answer 1

The Correct Option is B

Step 1: Set up remainder as ax + b and use x^2 = x + 3.
Let remainder \(R(x)=ax+b\). Since \(x^2=x+3\), compute successive powers: \(x^3=x^2\cdot x=(x+3)x=x^2+3x=4x+3\); \(x^4=x\cdot x^3=x(4x+3)=4x^2+3x=7x+12\); \(x^5=x\cdot x^4=7x^2+12x=19x+21\).

Step 2: Substitute into the polynomial and match.
\(2(19x+21)-3(7x+12)+5(4x+3)-3(x+3)+7x-9=38x+42-21x-36+20x+15-3x-9+7x-9=41x+3\). So \(a=41, b=3\), remainder \(=41x+3\). \[ \boxed{41x+3} \]