# Edexcel A Level Further Maths: Core Pure:复习笔记3.1.2 Linear Transformations of Roots

### 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

#### Exam Tip

• 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

#### Worked Example

The cubic equation  has roots  Find a polynomial equation with roots:

a)

b)