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

OCR A-Level Maths past-paper questions on 1.02.6 Polynomials and the factor theorem, 2018–2025, with mark schemes and examiner’s reports where available. Add MadAsMaths questions to your practice, or browse them separately by topic and difficulty.

All 17 questions on Polynomials and the factor theorem, as a list

June 2025, AS Paper 2, Q4 (10 marks): 2 4 The cubic polynomial x 2 kx 4 x + , where k is a constant, is denoted by ( )x k f . - + It is given that f ( ) 2 16 . l = (a) Show that k = 3. [4] For the r
June 2024, Paper 1, Q6 (8 marks): In this question you must show detailed reasoning. 3 2 The cubic polynomial ( )x f is defined by ( )x f 4 x 25 x 58 x 16 . = - - + 1 (a) Show that x is a root o
June 2024, Paper 2, Q3 (6 marks): 2 3 The function f is defined by ( ) f x x x 5 x - . 3 = - - (a) Show that ( x 3 ) is a factor of ( ) f x . [1] - (b) Factorise ( ) f x completely. [2] Three st
June 2024, Paper 2, Q6 (10 marks): In this question you must show detailed reasoning. (a) (i) Use the formula for cos A ( B ) , and the double angle formulae, to show that + 3i cos 3 4 cos 3 cos
June 2024, Paper 3, Q1 (4 marks): Simplify each of the following. ^ h 3 2 3 1 (a) 2 a a - [2] # 4 2 4 x 9 - (b) ^ h [2] 2 2 x 5 x 12 2 ( x 3 ) + - +
June 2023, Paper 1, Q2 (8 marks): 1 a 2 (a) (i) Show that can be written in the form b , where a, b and c are + cx 3 2 x 3 2 x + - + constants to be determined. [2] 1 1 (ii) Hence solve the equa
June 2023, Paper 3, Q3 (8 marks): 3 The cubic polynomial ( ) f x is defined by ( )x f x px + , where p and q are constants. q = + (a) (i) Given that f 2 ( ) 13 , find the value of p. [2] l = (ii
June 2022, Paper 2, Q1 (8 marks): In this question you must show detailed reasoning. Solve the following equations. x x 1 - (a) x 0 [3] - = x 1 2 + + 8 7 (b) 1 0 [3] 6 - - = 3 x x 1 x 7 (c) 3 2
June 2022, AS Paper 1, Q2 (5 marks): In this question you must show detailed reasoning. ^ h 3 2 The cubic polynomial f(x) is defined by f x 5 x 4 x ax 2 , where a is a constant. = - + - You are giv
November 2021, Paper 1, Q4 (7 marks): In this question you must show detailed reasoning. 3 2 The cubic polynomial ( ) f x is defined by ( ) f x 2 x 3 x 11 x + . 6 = - - (a) Use the factor theorem to
November 2021, Paper 2, Q8 (6 marks): 8 The number K is defined by K n + , where n is an integer greater than 2. 1 = ^h h, find the constants b and c. ^ 3 2 (a) Given that n 1 n 1 n bn c [1] + / + +
November 2021, AS Paper 2, Q1 (2 marks): + 1 Given that ( x 2 ) is a factor of x 2 kx - , find the value of the constant k. 4 [2] -
November 2020, Paper 1, Q12 (9 marks): Find the general solution of the differential equation d y 3 2 ( 2 x 3 x 11 x 6 ) y ( 20 x 35 ) . - - + = - d x Give your answer in the form y = f(x). [9]
November 2020, Paper 2, Q2 (4 marks): 2 2 x x 7 x - . 6 + - 2 Simplify fully [4] 2 x x 2 - -
November 2020, AS Paper 2, Q4 (9 marks): In this question you must show detailed reasoning. 3 2 The cubic polynomial x 6 kx 57 x 20 is denoted by f (x). It is given that ( 2 x 1 ) is a factor of + + -
June 2019, Paper 2, Q7 (5 marks): In this question you must show detailed reasoning. 6 2 4 3 2 Use the substitution u x + to solve the equation x 36 x 12 x 7 x x 2 0 . [5] = + + + - =
June 2019, AS Paper 1, Q3 (10 marks): In this question you must show detailed reasoning. 3 2 (a) The polynomial ( )x f is defined by ( )x f 2 x 3 x 8 x + . 3 = + - (i) Show that ( ) f 1 0 . [1] = (i

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