O-LevelAdditional MathematicsFactors of polynomials4037/22 · May/June 2024 · Q47 Marks

The polynomial p is such that p(x) = 6x³ +x² - 12x+5.

(a)Find the remainder when p(x) is divided by x-2.[1]
(b)(i)Show that 2x – 1 is a factor of p(x).[1]
(b)(ii)Hence write p(x) as a product of linear factors.[3]
(b)(iii)Hence solve the equation 6 sin³ θ+ sin²θ - 12 sinθ+5 = 0 for 0° ≤ θ ≤ 90°.[2]

Mark scheme

Marking points from the official mark scheme (May/June 2024). The mark or mark type is shown in brackets.

[B1] 33
[B1] (Implicitly for showing p(1/2) = 0)
[M2] Finds the quadratic factor 3x² + 2x – 5
[M2] M1 for any two terms correct in 3x² + 2x - 5
[A1] (2x-1)(3x + 5)(x – 1) oe
[A1] If 0 scored then SC2 for justifying x – 1 as a factor and writing down (2x – 1)(3x + 5)(x – 1) without any incorrect work seen
[B2] [sin θ = 0.5] θ = 30 nfww
[B2] B1for sinθ = 0.5 or θ = 30
[B2] [sin θ=1] θ = 90 nfww
[B2] or B1 for sinθ = 1 nfww or θ= 90 nfww
[B2] and no value of θ from 3sinθ+5 = 0

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Topic: Factors of polynomials, O-Level Additional Mathematics (syllabus 4037). Structured question, 7 marks.
Source: O-Level Additional Mathematics, May/June 2024 Paper 2 Variant 2, question 4.

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