The polynomial 2x³ + ax² + bx + 6, where a and b are constants, is denoted by p(x). When p(x) is divided by (x + 2) the remainder is –38 and when p(x) is divided by (2x – 1) the remainder is 19/2. Find the values of a and b.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2023). The mark or mark type is shown in brackets.

1. [M1] Substitute x = -2 and equate the result to -38 or divide by x + 2 to obtain quadratic quotient, and equate constant remainder to -38.
2. [A1] Obtain a correct evaluated equation, e.g. 2(-2)³ + a(-2)² + b(-2) + 6 = -38, leading to -16 + 4a – 2b + 6 = -38 or 4a – 2b = -28.
3. [M1] Substitute x = 1/2 and equate the result to 19/2 or divide by 2x – 1 to obtain quadratic quotient, and equate constant remainder to 19/2.
4. [A1] Obtain a correct evaluated equation, e.g. 2(1/2)³ + a(1/2)² + b(1/2) + 6 = 19/2, leading to 2/8 + a/4 + b/2 + 6 = 19/2 or 1/4 + a/4 + b/2 = 19/2 - 6 = 7/2, so a/4 + b/2 = 13/4 or a + 2b = 13.
5. [A1] Obtain a = -3 and b = 8.