The cubic equation x³ +ax²+bx-40 = 0 has three positive integer roots. Two of the roots are 2 and 4. Find the other root and the value of each of the integers 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. B1 2 × 4 × p = 40 → p = 5 May be obtained later.
2. M1 (x-2)(x-4)(x-p) = 0 Factorise cubic
3. A1 a = -11 Expand and identify
4. A1 b = 38
5. OR
6. B1 2 × 4 × p = 40 → p = 5 May be obtained later.
7. M1 Obtain equations 4a + 2b = 32, 16a+4b-24 and attempt to solve
8. A1 a = -11
9. A1 b = 38
10. M1 Find x = -1 Trial value/s and finds a root or shows that (x + 1) or (x + 4) or (x – 10) divides into x³ - 5x² - 46x – 40.
11. A1 (x + 1)(x² – 6x – 40) (= 0) or (x + 4)(x² – 9x – 10)(= 0) or (x – 10)(x² + 5x + 4)(= 0) Factorise to give linear and quadratic factor
12. M1 (x + 1)(x+4)(x – 10) (= 0) Solve the quadratic to give 2 roots
13. A1 x = -1, -4, 10
14. OR
15. M1 Uses factor theorem to find a root (-1)³-5(-1)² – 46(-1) – 40 or -1-5+46 - 40 = 0 → x = -1 This may be awarded for x = −4 or x = 10.
16. M1 At least two more trials.
17. A1 (-4)³-5(-4)²-46(-4) – 40 or -64-80+184-40 = 0 → x = -4
18. A1 (10)³ – 5(10)² – 46(10) – 40 or 1000-500-460-40 = 0 → x = 10