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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.
Problem 1: The quartic equation z4+2z3+6z2+8z+8=0 has a root z=−1+i. Find all four roots.
Solution:
Since coefficients are real, z=−1−i is also a root.
Quadratic factor: (z−(−1+i))(z−(−1−i))=(z+1)2+1=z2+2z+2.
Divide z4+2z3+6z2+8z+8 by z2+2z+2:
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