The polynomial x⁴ + 3x³ + ax + b, where a and b are constants, is denoted by p(x). When p(x) is divided by x² + x − 1 the remainder is 2x + 3. Find the values of a and b.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2019). The mark or mark type is shown in brackets.

1. [M1] Commence division and reach partial quotient x² + kx
2. [A1] Obtain correct quotient x² + 2x−1
3. [M1] Set their linear remainder equal to 2x + 3 and solve for a or for b
4. [A1] Obtain answer a = −1
5. [A1] Obtain answer b = 4
6. [M1] State x⁴ + 3x³ + ax + b = (x² +x−1)(x² + Ax + B) + 2x + 3 and form and solve two equations in A and B
7. [A1] Obtain A = 2, B = −1
8. [M1] Form and solve equations for a or b
9. [A1] Obtain answer a = −1
10. [A1] Obtain answer b = 4
11. [M1] Use remainder theorem with x = (-1±√5)/2
12. [A1] Expand brackets and obtain exact equation for either root. Accept exact equivalent.
13. [M1] Solve simultaneous equations for a or b
14. [A1] Obtain answer a = −1 from exact working
15. [A1] Obtain answer b = 4 from exact working