Perform polynomial division of P(x)=2x^3-3x^2+x-5 by D(x)=x-2. What are the quotient Q(x) and remainder R?

• A. Q(x)=2x^2+x+3, R=1
• B. Q(x)=2x^2+x+2, R=-1
• C. Q(x)=2x^2+x+4, R=0
• D. Q(x)=2x^2+x+1, R=-1

Answer: (A)

When dividing a polynomial by a linear polynomial, you get a quotient and a constant remainder, and the relation P(x) = D(x)Q(x) + R must hold. For the divisor x − 2, the remainder is a constant and, by the remainder theorem, R equals P(2).

Compute using synthetic division with the root 2. Take the coefficients of P: 2, −3, 1, −5. Bring down the leading coefficient 2. Multiply by 2 to get 4, add to −3 to get 1. Multiply by 2 to get 2, add to 1 to get 3. Multiply by 2 to get 6, add to −5 to get 1. So the quotient has coefficients 2, 1, 3, giving Q(x) = 2x^2 + x + 3, and the remainder is R = 1.

As a quick check, P(2) = 2(8) − 3(4) + 2 − 5 = 1, which matches the remainder. The result is Q(x) = 2x^2 + x + 3 and R = 1.