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Complex roots of cubic and quartic equationsEdexcel International A Level Further Maths: Flashcards

What these 12 flashcards ask

  • What does the conjugate root theorem say?
  • Real quadratic factor from roots p\pm qi?
  • Quadratic factor from roots 1\pm2i?
  • Does every real cubic have a real root?
  • How can the roots of a real quartic be arranged?
  • State the factor theorem.
  • How do you divide a quartic by a quadratic factor?
  • Factorise z^3-5z^2+17z-13 given z=1 is a root.
  • Roots of x^4-6x^3+18x^2-30x+25=0?
  • x^4-4x^3+6x^2-4x-15 has roots 3 and -1. Quadratic factor?
  • Why must you check the coefficients are real?
  • Where do conjugate roots lie on an Argand diagram?

Exam questions on Complex roots of cubic and quartic equations

  1. The cubic equation z3−5z2+17z−13=0z^3-5z^2+17z-13=0.
    Given that z=1z=1 is a root, find the other two roots.2 marks
  2. The cubic equation z3+z2−z+15=0z^3+z^2-z+15=0, which has a root 1−2i1-2\mathrm{i}.
    Find the quadratic factor of the cubic with real coefficients.2 marks
  3. The quartic function g(x)=x4−6x3+18x2−30x+25\text{g}(x)=x^4-6x^3+18x^2-30x+25, which has a root x=1+2ix=1+2\mathrm{i}.
    Write down a second root of g(x)=0\text{g}(x)=0 and hence find a quadratic factor of g(x)\text{g}(x) with real coefficients.3 marks
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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).