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Home / A-Level / Further Maths / Core Pure 1 / Roots of Polynomials / Roots of a Quartic Equation

Roots of a Quartic Equation

The big rules! (Quartic)
  1. \(\alpha+\beta+\gamma+\delta=-\frac{b}{a}\)
  2. \(\alpha\beta+\alpha\gamma+\beta\gamma+\alpha\delta+\beta\delta+\gamma\delta=\frac{c}{a}\)
  3. \(\alpha\beta\gamma+\alpha\beta\delta+\alpha\gamma\delta+\beta\gamma\delta=-\frac{d}{a}\)
  4. \(\alpha\beta\gamma\delta=\frac{e}{a}\)
From First Principles

Not done because it would take too long. We will not be asked to prove this in an exam.

The equation \(x^4+2x^3+px^2+qx-60=0\) where x is complex and p and q are real. Roots are \(\alpha\), \(\beta\), \(\gamma\) and \(\delta\). Given that \(\gamma=-2+4i\) and \(\delta=\gamma^*\), show that \(\alpha+\beta-2=0\) and \(\alpha\beta+3=0\).
  2. \(\alpha+\beta+\gamma+\delta=-\frac{b}{a}\)
  3. \(\alpha+\beta-2+4i-2-4i=-\frac{2}{1}\)
  4. \(\alpha+\beta-4=-2\)
  5. \(\alpha+\beta-2=0\)
  7. \(\alpha\beta\gamma\delta=\frac{e}{a}\)
  8. \(\alpha\beta(-2+4i)(-2-4i)=-60\)
  9. \(\alpha\beta(20)=-60\)
  10. \(\alpha\beta=-3\)
  11. \(\alpha\beta+3=0\)

    Now find all roots of the equation and the values of p and q.
  12. \(\alpha=2-\beta\)
  13. \((2-\beta)\beta+3=0\)
  14. \(-\beta^2+2\beta+3=0\)
  15. \(\beta^2-2\beta-3=0\)
  16. \(\beta=3\) or \(\beta=-1\)
  17. \(\alpha=-1\) or \(\alpha=3\)

    Now to find p using the \(\frac{c}{a}\) proof:
  18. \(3(-1)+3(-2+4i)+3(-2-4i)-1(-2-4i)+(-2+4i)(-2-4i)=\frac{p}{1}\)
  19. \(\therefore p=9\)

    Now to find q using the \(-\frac{d}{a}\) proof:
  20. \(3(-1)(-2+4i)+3(-1)(-2-4i)-1(-2+4i)(-2-4i)+3(-2+4i)(-2-4i)=-\frac{q}{1}\)
  21. \(\therefore q=-52\)

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