The cubic polynomial f(x) is defined by f(x) = x³ + ax² + 14x + a + 1, where a is a constant. It is given that (x + 2) is a factor of f(x).

Mark scheme

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

1. [M1] Substitute x = -2 and equate to zero
2. [A1] Obtain -8 + 4a - 28 + a + 1 = 0 or equivalent and hence a = 7
3. [M1] Attempt either division by x + 2 and reach partial quotient x² + kx, where k is numeric or use of identity or inspection or synthetic division
4. [A1] Obtain quotient x² + 5x + 4 soi
5. [A1] Conclude with (x + 1)(x + 2)(x + 4)
6. [M1] State (2x+1)(2x + 2)(2x + 4) = 3(x + 1)(x + 2)(x+4)
7. [A1] Obtain x = -1 and x = -2
8. [M1] Cancel common factors to obtain linear equation or factorise to find corresponding factor
9. [A1] Obtain x = 8/5 or equivalent
10. Or
11. [M1] State (2x+1)(2x + 2)(2x + 4) = 3(x + 1)(x + 2)(x + 4) or (2x)³ + 7(2x)² + 14(2x) + 8 = 3(x³ + 7x² + 14x + 8)
12. [A1] Expand and simplify to obtain 5x³ + 7x² - 14x - 16 = 0
13. [M1] Attempt complete factorisation of cubic with leading term 5x³ (may make use of synthetic division)
14. [A1] Obtain (x+1)(x+2)(5x-8)=0 and conclude x= -1, x = -2, x = 8/5