CAIE A-Level Mathematics A2 3.9.3 The Result That Questions

Practise finding and verifying complex roots of polynomial equations.

Exam points

use known roots to find other roots of polynomial equations
verify complex roots and determine remaining factors or roots
apply conjugate-root relationships for polynomials with 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 $-1-\sqrt{7} \mathrm{i}$
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
$-1-\sqrt{7} \mathrm{i}$
Or state $(x-(-1+\sqrt{7} \mathrm{i}))(x-(-1-\sqrt{7} \mathrm{i}))(2 x-p)$

Obtain $x^{2}+2 x+8$
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
Needs to follow from the working.

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CAIE A-Level Mathematics A2 3.9.3 The Result That Questions question 1