QCEVault

Further complex numbers — Question 271

Original QCE Vault practice · 6 marks

Q271 · Practice questionTechnology-freeComplex familiar6 marks

QUESTION 271 (6 marks)

The polynomial P(z)=z3+az2+bz+2P(z)=z^3+az^2+bz+2 has a,b∈Ca,b\in\mathbb C. Its remainder on division by z−iz-i is 2−2i2-2i, and P(1)=0P(1)=0.
a)
Use the remainder theorem to determine a and b.
[3 marks]
b)
Factorise P completely over C\mathbb C. Explain why real-coefficient conjugate-root reasoning cannot be assumed.
[3 marks]
Question linkSyllabus coverage

Related questions

  1. Q148 · Original practice · 1 mark
    The four roots of z4=−16z^4=-16 are shown on the Argand plane. For each root, a new complex number is defined by
    w=z+4z.w=z+\frac{4}{z}.
    Which statement describes the set of distinct values of ww?
    Further complex numbers
  2. Q153 · Original practice · 1 mark
    The points AA, BB and CC represent the three roots of z3=−8iz^3=-8i on the Argand plane. The point PP represents z=2z=2. What is the exact value of PA×PB×PCPA\times PB\times PC?
    Further complex numbers
  3. Q173 · Original practice · 5 marks
    Four beacons in a light installation have complex coordinates given by the solutions of (z−1−i)4=16(z-1-i)^4=16. They are represented on the Argand plane. The dashed circle has centre C=1+iC=1+i.
    Further complex numbers
  4. Q174 · Original practice · 8 marks
    A nonzero complex number satisfies ∣z∣=1|z|=1. Define w=z+1/zw=z+1/z.
    Further complex numbers