A-LevelMathematicsAlgebra9709/21 · Oct/Nov 2023 · Q59 Marks

The polynomial p(x) is defined by p(x) = 6x³ + ax² + bx – 20, where a and b are constants. It is given that (x + 2) is a factor of p(x) and that the remainder is –11 when p(x) is divided by (x + 1).

(a)Find the values of a and b.[5]
(b)Hence factorise p(x), and determine the exact roots of the equation p(3x) = 0.[4]

Mark scheme

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

[*M1] Substitute x=-2 and equate to zero
[*M1] Substitute x=-1 and equate to –11
[A1] Obtain 4a-2b-68=0 and a-b-26=-11 or equivalents
[DM1] Solve a pair of relevant simultaneous linear equations to find a or b Dependent at least one M1 mark.
[A1] Obtain a=19 and b=4
[M1] Divide by x+2 at least as far as the x term or equivalent (inspection, ...).
[A1] Obtain (x+2)²(6x – 5)
[M1] Replace (or imply replacement of) x by 3x in factorised form
[A1] Obtain −2/3 and 5/18 and no others.

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Topic: Algebra, A-Level Mathematics (syllabus 9709). Structured question, 9 marks.
Source: A-Level Mathematics, Oct/Nov 2023 Paper 2 Variant 1, question 5.

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