Edexcel A-Level Mathematics AS Fp1 1 5 Finding Conjugate Complex Roots Questions

Practise Edexcel IAL FP1.1.5 by using conjugate roots of real polynomials, forming factors, and finding remaining roots or coefficients.

Exam points

write down the conjugate root when a complex root of a real polynomial is given
form a quadratic factor from a conjugate pair before solving the remaining real factor
compare coefficients after factorisation to determine constants in a cubic equation

[Maximum number: 4]

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

\mathrm{f}(z)=4 z^{3}+p z^{2}-24 z+108

where p is a constant.
Given that -3 is a root of the equation f(z)=0

using algebra, solve f(z)=0 completely, giving the roots in simplest form,

$4z^3-8z^2-24z+108=(z+3)\left(4z^2+\ldots z+36\right)$ $=(z+3)\left(4z^2-20z+36\right)$ $4z^2-20z+36=0 \Rightarrow z=\frac{20\pm\sqrt{400-4\times4\times36}}{8}=\ldots$
Roots are $-3,\;\frac{5\pm \mathrm{i}\sqrt{11}}{2}$
(4)

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Edexcel A-Level Mathematics AS Fp1 1 5 Finding Conjugate Complex Roots Questions question 1