O-LevelAdditional MathematicsFactors of polynomials4037/11 · May/June 2023 · Q29 Marks

2 The polynomial p is such that p(x) = ax³ +7x²+bx+c, where a, b and c are integers.

(a)Given that p''(½) = 32, show that a = 6.[2]
(b)Given that p(x) has a factor of 3x-4 and a remainder of 7 when divided by x+1, find the values of b and c.[4]
(c)Write p(x) in the form (3x-4)q(x), where q(x) is a quadratic factor.[2]
(d)Hence write p(x) as a product of linear factors with integer coefficients.[1]

Mark scheme

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

[2] M1 for attempt to differentiate twice and substitute x = 1/2
a = 6
[4] M1 Must have 3 terms. For use of x = 4/3 , at least once and equating to 0 with an attempt at simplification leading to an equation in b and c only
Allow one sign error.
M1 Must have 3 terms. For use of x = -1 and equating to 7 with an attempt at simplification leading to an equation in b and c only
M1 dep on both previous M marks and attempt to solve simultaneously to obtain both b and c
A1 for both
[2] B1 for two terms correct in the quadratic factor. Allow if seen as a quotient in long division.
For both marks, need to see both factors together.
(x - 4/3)(6x² - 15x + 6) from synthetic method gets 0 marks unless recovered.
[1] B1 Must be all integers

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