CAIE A-Level Further Math AS 1.1.2 Root Transformations Questions

Practise transforming polynomial roots and forming equations for transformed roots.

Exam points

form equations with transformed polynomial roots
calculate sums of powers for transformed roots
solve root transformation problems in context

[Maximum number: 3]

The cubic equation $27 x^{3}+18 x^{2}+6 x-1=0$ has roots $\alpha, \beta, \gamma$ .

Show that a cubic equation with roots $3 \alpha+1,3 \beta+1,3 \gamma+1$ is

y^{3}-y^{2}+y-2=0

The sum $(3 \alpha+1)^{n}+(3 \beta+1)^{n}+(3 \gamma+1)^{n}$ is denoted by $S_{n}$ .

$\begin{aligned} y=3 x+1 \Rightarrow x=\frac{1}{3}(y-1) \quad \Rightarrow 27\left(\frac{y-1}{3}\right)^{3}+18\left(\frac{y-1}{3}\right)^{2}+6\left(\frac{y-1}{3}\right)-1=0 \end{aligned}$
Substitutes.

$\begin{aligned} \Rightarrow(y-1)^{3}+2(y-1)^{2}+2(y-1)-1=0 \quad \Rightarrow y^{3}-3 y^{2}+3 y-1+2 y^{2}-4 y+2+2 y-2-1=0 \end{aligned}$
Expands.

$y^{3}-y^{2}+y-2=0$
AG.

3

All question bank results loaded

CAIE A-Level Further Math AS 1.1.2 Root Transformations Questions question 1