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

Practise using conjugate-pair roots of real polynomials to complete equations, find factors and determine unknown 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