Polynomials and the factor theorem: Edexcel A‑Level Maths questions

Edexcel A-Level Maths past-paper questions on 2.6 Polynomials and the factor theorem, 2010–2025, with mark schemes and examiner’s reports where available. Add MadAsMaths questions to your practice, or browse them separately by topic and difficulty.

All 21 questions on Polynomials and the factor theorem, as a list

June 2025, Pure 1, Q4 (5 marks): Given that • f(x) = 2x 3 + 3x 2 – 16x + 16 • f(–4) = 0 (a) write f(x) in the form (x + a) Q(x) where a is a constant and Q(x) is a quadratic expression. (3) (b)
June 2025, Pure 1, Q12 (8 marks): O Q C l x y Figure 3 Figure 3 shows a sketch of the curve C with equation y = 15 2 3 3 x x x ( )( ) x ≠ – 3 2 x ≠ 3 and the straight line l with equation y = 2x
June 2025, Pure 2, Q1 (3 marks): In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. Factorise completely 2x 3 – 24x 2 + 40
June 2025, AS Pure, Q6 (7 marks): f (x) = ax 3 + bx 2 + 18x + 9 where a and b are constants. Given that (x + 3) is a factor of f (x) (a) show that –3a + b = 5 (2) Given also that f′ (2) = 14 (b)
June 2024, Pure 1, Q1 (3 marks): g(x) = x x k x k 3 20 17 3 2 where k is a constant. Given that (x – 3) is a factor of g(x), find the value of k. (3)
June 2024, AS Pure, Q2 (8 marks): In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. f (x) = 2x 3 – 3ax 2 + bx + 8
June 2024, AS Pure, Q5 (9 marks): The curve C1 has equation y = 6 x + 3 (a) (i) Sketch C1 stating the coordinates of any points where the curve cuts the coordinate axes. (ii) State the equations
June 2023, Pure 1, Q2 (6 marks): In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. f(x) = 4x 3 + 5x 2 – 10x + 4a
June 2023, AS Pure, Q15 (7 marks): In this question you must show detailed reasoning. Solutions relying on calculator technology are not acceptable. The curve C1 has equation The curve C2 has equ
June 2022, Pure 1, Q2 (3 marks): f (x) = (x − 4)(x2 − 3x + k) − 42 where k is a constant Given that (x + 2) is a factor of f (x) , find the value of k. (3)
June 2022, Pure 1, Q11 (7 marks): y x C1 O C2 Figure 4 Figure 4 shows a sketch of part of the curve C1 with equation y = 2x3 + 10 x > 0 and part of the curve C2 with equation y = 42x − 15x2 − 7
June 2022, AS Pure, Q2 (7 marks): f(x) = 2x3 + 5x2 + 2x + 15 (a) Use the factor theorem to show that (x + 3) is a factor of f(x). (2) (b) Find the constants a, b and c such that f(x) = (x + 3)(a
June 2022, AS Pure, Q7 (7 marks): (a) Factorise completely 9x – x3 (2) The curve C has equation y = 9x – x3 (b) Sketch C showing the coordinates of the points at which the curve cuts the x‑axis.
November 2021, Pure 1, Q1 (3 marks): f (x) = ax3 + 10x2 − 3ax − 4 Given that (x − 1) is a factor of f (x), find the value of the constant a. You must make your method clear. (3)
November 2021, AS Pure, Q6 (6 marks): In this question you should show all stages of your working. Solutions relying on calculator technology are not acceptable. (a) Using algebra, find all solution
November 2021, AS Pure, Q16 (11 marks): The curve C has equation y = f(x) where f(x) = ax 3 + 15x 2 – 39x + b and a and b are constants. Given • the point (2, 10) lies on C • the gradient of the curve
November 2020, Pure 2, Q8 (6 marks): A curve C has equation y = f (x) Given that ● f ʹ (x) = 6x2 + ax − 23 where a is a constant ● the y intercept of C is −12 ● (x + 4) is a factor of f (x) find, i
November 2020, AS Pure, Q10 (10 marks): g(x) = 2x3 + x2 – 41x – 70 (a) Use the factor theorem to show that g(x) is divisible by (x – 5). (2) (b) Hence, showing all your working, write g(x) as a produc
June 2019, Pure 1, Q1 (3 marks): f(x) = 3x3 + 2ax2 – 4x + 5a Given that (x + 3) is a factor of f(x), find the value of the constant a. (3)
June 2019, AS Pure, Q11 (10 marks): f(x) = 2x3 – 13x2 + 8x + 48 (a) Prove that (x – 4) is a factor of f(x). (2) (b) Hence, using algebra, show that the equation f(x) = 0 has only two distinct root
June 2018, Pure 2, Q6 (6 marks): f (x) = −3x3 + 8x2 − 9x + 10, x ∈  (a) (i) Calculate f (2) (ii) Write f (x) as a product of two algebraic factors. (3) Using the answer to (a)(ii), (b) prove t

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