Applications and Problem Solving

This final lesson brings together all the complex number techniques covered in this course. We focus on exam-style problem solving, combining multiple ideas within a single question — exactly as you will encounter in the AQA Further Mathematics exam.

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Strategy for Complex Number Problems

1. Read the question carefully — identify what form is needed (Cartesian, polar, exponential).
2. Choose the right representation — Cartesian for addition/subtraction, polar for multiplication/powers/roots.
3. Show all working — examiners award method marks.
4. Check your answer — substitute back, verify modulus/argument, check conjugate pairs.

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Type 1: Polynomial Equations

Problem 1: The quartic equation $z^4 + 2z^3 + 6z^2 + 8z + 8 = 0$z4+2z3+6z2+8z+8=0 has a root $z = -1 + i$z=−1+i. Find all four roots.

Solution:

Since coefficients are real, $z = -1 - i$z=−1−i is also a root.

Quadratic factor: $(z - (-1 + i))(z - (-1 - i)) = (z + 1)^2 + 1 = z^2 + 2z + 2$(z−(−1+i))(z−(−1−i))=(z+1)2+1=z2+2z+2.

Divide $z^4 + 2z^3 + 6z^2 + 8z + 8$z4+2z3+6z2+8z+8 by $z^2 + 2z + 2$z2+2z+2: