CAIE A-Level Mathematics 3.9.3 The result that

Practise using conjugate root pairs in real-coefficient polynomial equations to find missing roots, factors and coefficient values.

Exam points

state the conjugate root when a non-real root of a real polynomial is given
build a real quadratic factor from conjugate roots before finding remaining roots
use sums and products of conjugate roots to calculate real coefficients

[Maximum number: 4]

The complex number $-1+\sqrt{7} \mathrm{i}$ is denoted by u. It is given that u is a root of the equation

2 x^{3}+3 x^{2}+14 x+k=0,

where k is a real constant.

Find the other two roots of the equation.

State answer $-1-\sqrt{7} \mathrm{i}$

B1

Can be seen simply stated on its own, or in a list of roots.

Marking guidance:

Allow if stated clearly in part 10(a).

Carry out a method for finding a quadratic factor with zeros $-1+\sqrt{7} \mathrm{i}$ and
$-1-\sqrt{7} \mathrm{i}$

M1

Or state $(x-(-1+\sqrt{7} \mathrm{i}))(x-(-1-\sqrt{7} \mathrm{i}))(2 x-p)$

Obtain $x^{2}+2 x+8$

A1

Or obtain $(-1+\sqrt{7} \mathrm{i})(-1-\sqrt{7} \mathrm{i}) p=-8$
Or obtain $(-1+\sqrt{7} \mathrm{i})+(-1-\sqrt{7} \mathrm{i})+\frac{p}{2}=-\frac{3}{2}$

Obtain root $x=\frac{1}{2}$ , or equivalent, via division or inspection

A1

Needs to follow from the working.

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3.9.3—The result that question 1