Find the quotient and remainder when 2x⁴ – 27 is divided by x² + x + 3.

Mark scheme

Marking points from the official mark scheme (May/June 2023). The mark or mark type is shown in brackets.

1. [M1] Divide to obtain quotient 2x² +2x+k (k≠ 0); Obtain result in answer column, together with a linear polynomial or a constant as remainder. If correct: (2x^4 - 27) / (x^2 + x + 3) = 2x^2 - 2x - 4 + (10x - 15) / (x^2 + x + 3)
2. [A1] Obtain [quotient] 2x²-2x-4; Allow unless quotient and remainder interchanged, then A0 A1.
3. [A1] Obtain [remainder] 10x-15; Allow (x² + x + 3)(2x² – 2x – 4) + 10x – 15. Alternative Method: [M1] Expand (x² + x + 3)(Ax² + Bx +C) + (Dx+E) and reach A = 2, B = ± 2, C=k; Solve all 3 equations for A, B and C, allow sign errors in establishing equations and in solving. If correct, A = 2, A + B = 0, 3A + B + C = 0, 3B + C + D = 0, 3C + E = - 27. Obtain result in answer column, together with a linear polynomial or a constant as remainder. [A1] Obtain [quotient] 2x² – 2x-4; Allow unless quotient and remainder interchanged, then A0 A1. [A1] Obtain [remainder] 10x-15; Allow (x² + x + 3) (2x² – 2x – 4) + 10x – 15.