O-LevelAdditional MathematicsFactors of polynomials4037/11 · May/June 2021 · Q38 Marks

The polynomial p(x) = ax³-9x²+bx-6, where a and b are constants, has a factor of x-2. The polynomial has a remainder of 66 when divided by x-3.

(a)Find the value of a and of b.[4]
(b)Using your values of a and b, show that p(x) = (x-2)q(x), where q(x) is a quadratic factor to be found.[2]
(c)Hence show that the equation p(x) = 0 has only one real solution.[2]

Mark scheme

Marking points from the official mark scheme (May/June 2021). The mark or mark type is shown in brackets.

[4] B1 p(2): 8a-36+2b-6=0
[4] B1 p(3): 27a-81+3b-6=66
[4] M1 Dep on at least one of the previous B marks, for attempt to solve their equations and obtain a solution for both a and b
[4] A1 For both
[2] M1 for attempt at quadratic factor either by observation to obtain 6x² + px +3 or by algebraic long division to obtain at least 6x² + 3x...
[2] A1 all correct
[2] M1 For calculation of discriminant and confirmation that it is <0
[2] A1 For a correct conclusion from correct work.

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Topic: Factors of polynomials, O-Level Additional Mathematics (syllabus 4037). Structured question, 8 marks.
Source: O-Level Additional Mathematics, May/June 2021 Paper 1 Variant 1, question 3.

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