Roots of polynomial equations — AS & A Level Further Mathematics (9231) questions by topic
Vieta's formulas and transformed-root equations run through this topic: stating a symmetric function of a cubic's roots in terms of p, showing a transformed cubic with roots 3alpha + 1 and so on, and finding the sum of the squares of a quartic's four roots from given symmetric-function data.
Find, show, state, deduce and calculate lead the command words across 15 questions from 2023 to 2025 papers, worth 6 to 10 marks, where a sign error in a Vieta substitution is the single most common way to lose marks. Checking each symmetric-function substitution with instant marking flags a sign slip on the sum of roots before it carries through several subsequent lines of algebra.
What examiners look for
- A recurring criticism is sign errors when applying Vieta's formulas, such as giving p/2 instead of -p/2 by forgetting the coefficient of x is -p, not p.
- Examiners note dividing by the wrong leading coefficient, using 1 instead of the correct value of 2, as a common slip.
- A common mistake substituted +1/2 instead of -1/2 for the sum of roots, producing +5/4 instead of -5/4.
- Reports flag failing to spot the factorisation of alpha*beta*gamma out of all three terms, instead trying to expand or evaluate term by term, which usually went nowhere in the time available.
- Strong scripts identified the coefficient of x as -p, divided by the leading coefficient of 2 in one line, and quoted the symmetric-function values explicitly so any slip stayed visible.
May–June 2025, Paper 11
The cubic equation x^3 + 2x + 1 = 0 has roots alpha, beta, gamma. Find a cubic equation whose roots are alpha^3 - 1, beta^3 - 1, gamma^3 - 1.
Answer this question and get it marked →May–June 2025, Paper 12
The cubic equation x^3 + 2x + 1 = 0 has roots alpha, beta, gamma. Find a cubic equation whose roots are alpha^3 - 1, beta^3 - 1, gamma^3 - 1.
Answer this question and get it marked →May–June 2025, Paper 13
The quartic equation x^4 + 7x^2 + 3x + 22 = 0 has roots alpha, beta, gamma, delta. Find the value of alpha^2 + beta^2 + gamma^2 + delta^2.
Answer this question and get it marked →May–June 2024, Paper 11
The cubic equation 2x^3 + x^2 - px - 5 = 0, where p is a positive constant, has roots alpha, beta, gamma. State, in terms of p, the value of alphabeta + betagamma + gammaalpha.
Answer this question and get it marked →May–June 2024, Paper 12
The cubic equation 2x^3 + x^2 - px - 5 = 0, where p is a positive constant, has roots alpha, beta, gamma. State, in terms of p, the value of alphabeta + betagamma + gammaalpha.
Answer this question and get it marked →May–June 2024, Paper 13
The cubic equation x^3 + 2x^2 + 3x + 1 = 0 has roots alpha, beta, gamma. Find a cubic equation whose roots are alpha^2 + 1, beta^2 + 1, gamma^2 + 1.
Answer this question and get it marked →October–November 2024, Paper 11
The quartic equation x^4 + 2x^3 - 1 = 0 has roots alpha, beta, gamma, delta. Find a quartic equation whose roots are alpha^4, beta^4, gamma^4, delta^4 and state the value of…
Answer this question and get it marked →October–November 2024, Paper 12
It is given that alpha + beta + gamma + delta = 2, alpha^2 + beta^2 + gamma^2 + delta^2 = 3, alpha^3 + beta^3 + gamma^3 + delta^3 = 4. Find the value of alphabeta + alphagamma +…
Answer this question and get it marked →October–November 2024, Paper 13
The quartic equation x^4 + 2x^3 - 1 = 0 has roots alpha, beta, gamma, delta. Find a quartic equation whose roots are alpha^4, beta^4, gamma^4, delta^4 and state the value of…
Answer this question and get it marked →May–June 2023, Paper 11
The cubic equation x^3 + 4x^2 + 6x + 1 = 0 has roots alpha, beta, gamma. Find the value of alpha^2 + beta^2 + gamma^2.
Answer this question and get it marked →May–June 2023, Paper 12
The cubic equation x^3 + 4x^2 + 6x + 1 = 0 has roots alpha, beta, gamma. Find the value of alpha^2 + beta^2 + gamma^2.
Answer this question and get it marked →May–June 2023, Paper 13
The equation x^4 - x^2 + 2x + 5 = 0 has roots alpha, beta, gamma, delta. Find a quartic equation whose roots are alpha^2, beta^2, gamma^2, delta^2 and state the value of alpha^2 +…
Answer this question and get it marked →October–November 2023, Paper 11
The quartic equation x^4 + b x^3 + c x^2 + d x - 2 = 0 has roots alpha, beta, gamma, delta. It is given that alpha + beta + gamma + delta = 3, alpha^2 + beta^2 + gamma^2 + delta^2…
Answer this question and get it marked →October–November 2023, Paper 12
The cubic equation 27x^3 + 18x^2 + 6x - 1 = 0 has roots alpha, beta, gamma. Show that a cubic equation with roots 3alpha + 1, 3beta + 1, 3gamma + 1 is y^3 - y^2 + y - 2 = 0.
Answer this question and get it marked →October–November 2023, Paper 13
The quartic equation x^4 + bx^3 + cx^2 + dx - 2 = 0 has roots alpha, beta, gamma, delta. It is given that alpha + beta + gamma + delta = 3, alpha^2 + beta^2 + gamma^2 + delta^2 =…
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