A-LevelMathematicsAlgebra9709/31 · May/June 2018 · Q45 Marks

The polynomial x⁴ + 2x³ + ax + b, where a and b are constants, is divisible by x² − x + 1. Find the values of a and b.

(a)The polynomial x⁴ + 2x³ + ax + b, where a and b are constants, is divisible by x² − x + 1. Find the values of a and b.[5]

Mark scheme

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

[M1] EITHER: Commence division by x² − x +1 and reach a partial quotient of the form x² + kx
[A1] Obtain quotient x² +3x+2
[M1] Either Set remainder identically equal to zero and solve for a or for b, or multiply given divisor and found quotient and obtain a or b
[A1] Obtain a = 1
[A1] Obtain b = 2
[M1] OR: Assume an unknown factor x² + Bx + C and obtain an equation in B and/or C
[A1] Obtain B = 3 and A = 2
[M1] Either Use equations to obtain a or b or multiply given divisor and found factor to obtain a or b
[A1] Obtain a = 1
[A1] Obtain b = 2

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Topic: Algebra, A-Level Mathematics (syllabus 9709). Structured question, 5 marks.
Source: A-Level Mathematics, May/June 2018 Paper 3 Variant 1, question 4.

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