Do not use a calculator in this question. It is given that x + 4 is a factor of p(x) = 2x³ + 3x² + ax - 12. When p(x) is divided by x - 1 the remainder is b.

Mark scheme

Marking points from the official mark scheme (May/June 2018). The mark or mark type is shown in brackets.

1. [1] (i) 2(-4)³ + 3(-4)² - 4a - 12 = 0 with one correct interim step leading to a = -23
2. [1] (i) Note: = 0 must be seen or may be implied by e.g. -92 = 4a or 92 = -4a or convincingly showing that 2(-4)³ + 3(-4)² - 4(-23) - 12 = 0 or correct synthetic division at least as far as -4 | 2 3 a -12 | -8 20 -4a-80 ---------------------- 2 -5 a+20 0 then a = -23 or correct long division to, e.g. verify -23, at least as far as...
3. [1] (i) p(1) = 2+3-23-12
4. [1] (i) b = -30
5. [2] (ii) finds a correct quadratic factor e.g. (2x² - 5x - 3)
6. [1] (ii) B1 for quadratic factor with 2 correct terms OR B1 for finding (x – 3) using factor theorem B1 for convincingly finding (2x + 1) as third factor
7. [1] (ii) Product of three linear factors (2x + 1)(x – 3)(x + 4)
8. [1] (ii) x = -1/2, x = 3, x = -4 nfww
9. [1] (ii) If M0 then SC1 if quadratic factorised correctly but does not show full factorisation but does give all 3 solutions correctly