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MathidiaMathematics Academy
Polynomial roots
Lesson 12 of 75
Easy practice

Transformed roots

Objective. Form equations whose roots are shifted, scaled or reciprocated.

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  2. 2. Worked example
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Learn: the key idea

Given a polynomial with known roots, a new polynomial can be formed whose roots are a transformation of the original roots, such as each root shifted, scaled or reciprocated. The standard method substitutes an expression for the old variable in terms of the new one into the original equation. If y = x + k, substitute x = y − k into the original equation; if y = 1/x, substitute x = 1/y and clear fractions. For example, if x³−2x+1=0 has roots α, the equation with roots 2α is found by substituting x = y/2.

Worked example

The equation x² − 3x + 2 = 0 has roots α and β. Find the equation whose roots are α + 1 and β + 1.

3 … 2
Illustrated card showing 3 … 2
  1. 1Let y = x + 1, so x = y − 1
  2. 2Substitute into x² − 3x + 2 = 0: (y−1)² − 3(y−1) + 2 = 0
  3. 3Expand: y² − 2y + 1 − 3y + 3 + 2 = 0
  4. 4Simplify: y² − 5y + 6 = 0

Answer: y² − 5y + 6 = 0

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Common mistake. Students substitute the transformation the wrong way round, e.g. putting y = 1/x then substituting y = 1/x instead of x = 1/y. Always isolate the old variable x in terms of the new variable y before substituting.

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