Linear Transformations of Roots

What is a Linear Transformation?

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

How do I find the new transformed polynomial?

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

Examiner 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 $x^{3} - 7 x^{2} + 2 x + 40 = 0$ 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)

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What is a Linear Transformation?

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1. Complex Numbers

1.1 Complex Numbers & Argand Diagrams

1.1.1 Introduction to Complex Numbers

1.1.2 Solving Equations with Complex Roots

1.1.3 Modulus & Argument

1.1.4 Modulus-Argument Form

1.1.5 Loci in Argand Diagrams

1.1.6 Regions in Argand Diagrams

1.2 Exponential Form & de Moivre's Theorem

1.2.1 Exponential Form

1.2.2 de Moivre's Theorem

1.2.3 Applications of de Moivre's Theorem

1.2.4 Roots of Complex Numbers

2. Matrices

2.1 Properties of Matrices

2.1.1 Introduction to Matrices

2.1.2 Determinants of Matrices

2.1.3 Inverses of Matrices

2.2 Transformations using Matrices

2.2.1 Transformations using a Matrix

2.2.2 Geometric Transformations with Matrices

2.2.3 Invariant Points & Lines

3. Further Algebra & Functions

3.1 Roots of Polynomials

3.1.1 Roots of Polynomials

3.1.2 Linear Transformations of Roots

3.2 Series

3.2.1 Sums of Integers, Squares & Cubes

3.2.2 Method of Differences

3.3 Maclaurin Series

3.3.1 Maclaurin Series

4. Hyperbolic Functions

4.1 Hyperbolic Functions

4.1.1 Hyperbolic Functions & Graphs

4.1.2 Logarithmic Forms of Inverse Hyperbolic Functions

4.1.3 Hyperbolic Identities & Equations

4.1.4 Differentiating & Integrating Hyperbolic Functions

5. Further Calculus

5.1 Volumes of Revolution

5.1.1 Volumes of Revolution

5.1.2 Modelling with Volumes of Revolution

5.2 Methods in Calculus

5.2.1 Improper Integrals

5.2.2 Mean Value of a Function

5.2.3 Integrating with Partial Fractions

5.2.4 Calculus involving Inverse Trig

5.2.5 Integration by Substitution

6. Further Vectors

6.1 Vector Lines

6.1.1 Equations of Lines in 3D

6.1.2 Pairs of Lines in 3D

6.1.3 Angle between Lines

6.1.4 Shortest Distances - Lines

6.2 Vector Planes

6.2.1 Equations of planes

6.2.2 Combinations of Lines & Planes

6.2.3 Combinations of Planes

6.2.4 Shortest Distances - Planes

7. Polar Coordinates

7.1 Polar Coordinates

7.1.1 Polar Coordinates

7.1.2 Calculus with Polar Coordinates

8. Differential Equations

8.1 First Order Differential Equations

8.1.1 Intro to Differential Equations

8.1.2 Solving First Order Differential Equations

8.1.3 Modelling using First Order Differential Equations

8.2 Second Order Differential Equations

8.2.1 Solving Second Order Differential Equations

8.2.2 Coupled First Order Linear Equations

8.3 Simple Harmonic Motion

8.3.1 Simple Harmonic Motion

8.3.2 Damped or Forced Harmonic Motion