The polynomial 2x⁴ + ax³ + bx − 1, where a and b are constants, is denoted by p(x). When p(x) is divided by x² – x + 1 the remainder is 3x + 2. Find the values of a and b.

Mark scheme

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

1. [M1] Commence division and reach partial quotient 2x² + (a + 2)x
2. [A1] Obtain correct quotient 2x² + (a + 2)x + a
3. [M1] Set their linear remainder equal to part of “3x + 2" and solve for a or for b
4. [A1] Obtain answer a = -3
5. [A1] Obtain answer b = 5
6. Alternative method for Question 3
7. [M1] State 2x⁴ + ax³ +0x² +bx-1=(x²-x+1)(2x² + Ax + B)+3x+2 and form and solve equation(s) to obtain A or B
8. [A1] Obtain A = -1, B = −3
9. [M1] Form and solve equations for a or for b
10. [A1] Obtain answer a = -3
11. [A1] Obtain answer b = 5
12. Alternative method for Question 3
13. [M1] Use remainder theorem with x = (1±√-3)/2 or x = (1±i√3)/2
14. [A1] Expand brackets and obtain exact equation for either root. Accept exact equivalent.
15. [M1] Solve simultaneous equations, or single equation, for a or for b
16. [A1] Obtain answer a = –3 from exact working
17. [A1] Obtain answer b = 5 from exact working