Linear Transformations of Roots (Edexcel A Level Further Maths): Revision Note

Linear Transformations of Roots

What is a Linear Transformation?

• A linear transformation can be expressed as  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

How do I find the new transformed polynomial?

• STEP 1 Rewrite the transformation  as 

• STEP 2 Make the substitution  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