The polynomial 4x³ + ax² +5x+b, where a and b are constants, is denoted by p(x). It is given that (2x+1) is a factor of p(x). When p(x) is divided by (x-4) the remainder is equal to 3 times the remainder when p(x) is divided by (x-2). Find the values of a and b.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2024). The mark or mark type is shown in brackets.

1. [M1] Substitute x = -½ and equate the result to zero
2. [A1] Obtain a correct equation, e.g. -½ + ¼a - 5/2 + b = 0 (¼a + b = 3) Any equivalent form.
3. [M1] Substitute x = 2 and x = 4 and use p(4) = 3p(2) If using long division, M1 is for correct use of two constant remainders. Condone if 3 is on the wrong side.
4. [A1] Obtain a correct equation, e.g. 3(32 + 4a + 10 + b) = 256 +16a + 20+ b (−2a + b = 75) Any equivalent form.
5. [A1] Obtain a = -32 and b = 11