plumiRoots of polynomials: AQA A‑Level Further Maths questions
AQA A-Level Further Maths past-paper questions on D.1 Roots of polynomials, 2019–2025, with mark schemes and examiner’s reports where available. Add MadAsMaths questions to your practice, or browse them separately by topic and difficulty.
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All 15 questions on Roots of polynomials, as a list
- June 2025, Paper 1, Q7 (4 marks): The cubic equation + − + = 3 2 5 4 3 0 z z z has roots α, β and γ Find an equation, with integer coefficients, that has roots − 2 1 α , − 2 1 β and − 2 1 γ [4 m
- June 2025, AS Paper 1, Q5 (2 marks): The quartic equation 3x4 + x2 – 5x – 11 = 0 has roots α, β, γ and δ 5 (a) Write down the value of αβγδ [1 mark] 5 (b) Write down the value of αβγ + αβδ + αγδ +
- June 2024, Paper 1, Q1 (1 mark): The roots of the equation 20x3 – 16x2 – 4x + 7 = 0 are α, β and γ Find the value of αβ + βγ + γα Circle your answer. [1 mark] _ – 4 5 – 1 5 1 5 4 5
- June 2024, Paper 2, Q6 (3 marks): The cubic equation x3 + 5x2 – 4x + 2 = 0 has roots α, β and γ Find a cubic equation, with integer coefficients, whose roots are 3α, 3β and 3γ [3 marks]
- June 2024, AS Paper 1, Q13 (5 marks): The cubic equation x3 – x – 7 = 0 has roots α, β and γ The cubic equation p(x) = 0 has roots α – 1, β – 1 and γ – 1 The coefficient of x3 in p(x) is 1 13 (a) De
- June 2023, Paper 2, Q13 (11 marks): The quadratic equation z2 − 5z + 8 = 0 has roots a and b 13 (a) Write down the value of a + b and the value of ab [2 marks] 13 (b) Without finding the value of
- June 2023, AS Paper 1, Q4 (1 mark): The roots of the equation 5x3 + 2x2 – 3x + p = 0 are α, β and γ Given that p is a constant, state the value of αβ + βγ + γα Circle your answer. [1 mark] – 3 5 –
- June 2022, Paper 2, Q3 (1 mark): The roots of the equation x2 − px − 6 = 0 are a and b Find a2 + b2 in terms of p Circle your answer. [1 mark] p2 − 6 p2 + 6 p2 − 12 p2 + 12
- June 2022, AS Paper 1, Q2 (1 mark): The quadratic equation x2 + px + q = 0 has roots a and b Which of the following is equal to ab ? Circle your answer. [1 mark] p −p q −q 1 0 0 2 3
- November 2021, Paper 2, Q5 (5 marks): The equation z3 + 2z2 − 5z − 3 = 0 has roots a, b and g Find a cubic equation with roots 1 2 a − 1, 1 2 b − 1 and 1 2 g − 1 [5 marks]
- November 2020, Paper 1, Q3 (1 mark): The quadratic equation ax2 + bx + c = 0 (a, b, c 2 R) has real roots a and b. One of the four statements below is incorrect. Which statement is incorrect? Tick
- November 2020, Paper 1, Q8 (6 marks): The three roots of the equation 4x3 − 12x2 − 13x + k = 0 where k is a constant, form an arithmetic sequence. Find the roots of the equation. [6 marks]
- November 2020, AS Paper 1, Q9 (8 marks): The quadratic equation 2x2 + px + 3 = 0 has two roots, a and b, where a > b. 9 (a) (i) Write down the value of ab. [1 mark] 9 (a) (ii) Express a + b in terms of
- June 2019, Paper 1, Q13 (14 marks): The equation z3 + kz2 + 9 = 0 has roots a, b and g. 13 (a) (i) Show that a2 + b2 + g2 = k 2 [3 marks] 13 (a) (ii) Show that a2b2 + b2g2 + g2a2 = −18k [4 marks]
- June 2019, AS Paper 1, Q14 (7 marks): The graph of y = x3 − 3x is shown below. x y y = x3 – 3x O The two stationary points have x-coordinates of −1 and 1 The cubic equation x3 − 3x + p = 0 where p i
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