The polynomial p(x) is defined by p(x) = 2x³ + ax² – 3x – 4, where a is a constant. It is given that (x - 4) is a factor of p(x).

Mark scheme

Marking points from the official mark scheme (May/June 2022). The mark or mark type is shown in brackets.

1. [M1] Substitute x = 4, equate to zero and attempt solution
2. [A1] Obtain a = -7
3. [M1] Divide by x-4 at least as far as the x term or use of identity or by inspection
4. [A1] Obtain 2x² + x +1 and conclude (x-4)(2x² +x+1)
5. [M1] Divide by x-4 at least as far as the x term
6. [M1] Equate the remainder to zero
7. [A1] Obtain a = -7
8. [A1] Obtain 2x² +x+1 and conclude (x-4)(2x² + x + 1)
9. [M1] Apply logarithms and use power law for e³y = 4
10. [A1] Obtain ln 4 or exact equivalent
11. [B1] Use discriminant [=1-8=-7] or equivalent to show no other roots AG - necessary detail needed