Complex numbers — Question 7
Original QCE Vault practice · 5 marks
Q7 · Practice questionComplex familiar5 marks
QUESTION 7 (5 marks)
The complex number z satisfies
z4+4=0.a)Determine the four values of z in Cartesian form.
b)Hence factorise z4+4 into two quadratic factors with real coefficients.
WORKED SOLUTION
Practice marking scheme
5 marksANSWERz=1+i,−1+i,−1−i,1−i;z4+4=(z2−2z+2)(z2+2z+2) Sample response and mark allocation
z4=−4=4cis(π+2kπ).
Expresses the equation in polar form.[1 mark]
z=2cis(π/4+kπ/2), k=0,1,2,3.
Determines the four polar-form roots.[1 mark]
z=1+i,−1+i,−1−i,1−i.
Expresses all roots in Cartesian form.[1 mark]
(z−(1+i))(z−(1−i))=z2−2z+2 and
(z−(−1+i))(z−(−1−i))=z2+2z+2.
Groups conjugate roots into real quadratic factors.[1 mark]
z4+4=(z2−2z+2)(z2+2z+2).
States the required factorisation.[1 mark]
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