linear-transformations-of-roots

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Linear transformations of roots

What is a linear transformation?

A linear transformation can be expressed as $w equals p x plus q$ 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 $w equals p x plus q$ as $x equals fraction numerator w minus q over denominator p end fraction$
STEP 2 Make the substitution $x equals fraction numerator w minus q over denominator p end fraction$ 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 Tips and Tricks

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 <img alt=”x cubed minus 7 x squared plus 2 x plus 40 equals 0″ data-mathml='<math ><semantics><mrow><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mn>7</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>40</mn><mo>=</mo><mn>0</mn></mrow><annotation encoding=”application/vnd.wiris.mtweb-params+json”>{“language”:”en”,”fontFamily”:”Times New Roman”,”fontSize”:”18″}</annotation></semantics></math>’ height=”23″ role=”math” 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Further Maths: Core Pure -Edexcel-A Level

linear-transformations-of-roots