Linear Transformations of Roots

A linear transformation can be expressed as $w = p x + q$ where:
- w is the new root
- x is the original root
- p and q are constants
If you perform a linear transformation on a polynomial equation, the transformed equation’s roots will be linked directly to the original roots
You can think of this as a translation and/or a stretch of the polynomial and its roots

STEP 1
Rewrite the transformation $w = p x + q$ as $x = \frac{w - q}{p}$
STEP 2
Make the substitution $x = \frac{w - q}{p}$ into the original polynomial
STEP 3
Expand and multiply by a constant to make all the coefficients integers (if necessary or desirable) and swap the w back for an x in your final answer
Remember that your solution is not unique. Multiplying the entire polynomial by a constant will produce a different polynomial, but will not affect the roots

Check the question to see if you are required to expand and simplify your final answer or not, as it can be time consuming!
If you are required to expand and simplify, make use of the binomial expansion to make the process much quicker
Use your calculator’s polynomial solver to check the solutions of the original equation and your new equation, to make sure they are related to each other as described

The cubic equation $x^{3} - 7 x^{2} + 2 x + 40 = 0$ has roots $α, β, γ .$ Find a polynomial equation with roots:

3 α, 3 β, 3 γ

3-1-2-edx-a-fm-we1a-soltn

(2 α - 1), (2 β - 1), (2 γ - 1)

3-1-2-edx-a-fm-we1b-soltn

Linear Transformations of Roots (Edexcel A Level Further Maths: Core Pure)