Roots of polynomial equations: past paper questions

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• Question 4May/June 2026, paper 11Structured10 marks

  The cubic equation $5x^3-2x+1=0$ has roots $\alpha ,\beta ,\gamma$. Find a cubic equation whose roots are $\frac{1}{{\alpha }^2},\frac{1}{{\beta }^2},\frac{1}{{\gamma }^2}$.

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• Question 4Oct/Nov 2025, paper 11Structured8 marks

  The quartic equation $x^4+x^3+x^2+x+1=0$ has roots $\alpha ,\beta ,\gamma ,\delta$. Show that a quartic equation with roots $2\alpha +1,2\beta +1,2\gamma +1,2\delta +1$ is $$y^4-2y^3+4y^2+2y+11=0.$$

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• Question 3Oct/Nov 2024, paper 11Structured10 marks

  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 ${\alpha }^4+{\beta }^4+{\gamma }^4+{\delta }^4$.

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• Question 2Oct/Nov 2025, paper 12Structured8 marks

  The cubic equation $x^3+b{x}^2+cx+d=0$, where $b$, $c$ and $d$ are constants, has roots $\alpha$, $\beta$ and $\gamma$.

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• Question 4Oct/Nov 2023, paper 12Structured10 marks

  The cubic equation $27x^3+18x^2+6x-1=0$ has roots $\alpha$, $\beta$, $\gamma$. Show that a cubic equation with roots $3\alpha +1$, $3\beta +1$, $3\gamma +1$ is $$y^3-y^2+y-2=0.$$

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• Question 2May/June 2025, paper 11Structured7 marks

  The cubic equation $x^3+2x+1=0$ has roots $\alpha ,\beta ,\gamma$. Find a cubic equation whose roots are ${\alpha }^3-1$, ${\beta }^3-1$, ${\gamma }^3-1$.

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• Question 1May/June 2024, paper 11Structured6 marks

  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 $\alpha \beta +\beta \gamma +\gamma \alpha$.

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• Question 2May/June 2023, paper 11Structured8 marks

  The cubic equation $x^3+4x^2+6x+1=0$ has roots $\alpha ,\beta ,\gamma$. Find the value of ${\alpha }^2+{\beta }^2+{\gamma }^2$.

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• Question 3May/June 2025, paper 13Structured9 marks

  The quartic equation $x^4+7x^2+3x+22=0$ has roots $\alpha ,\beta ,\gamma ,\delta$. Find the value of ${\alpha }^2+{\beta }^2+{\gamma }^2+{\delta }^2$.

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• Question 2May/June 2024, paper 13Structured7 marks

  The cubic equation $x^3+2x^2+3x+1=0$ has roots $\alpha ,\beta ,\gamma$. Find a cubic equation whose roots are ${\alpha }^2+1,{\beta }^2+1,{\gamma }^2+1$.

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• Question 3May/June 2023, paper 13Structured9 marks

  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 ${\alpha }^2+{\beta }^2+{\gamma }^2+{\delta }^2$.

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