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MathidiaMathematics Academy
Polynomial roots
Lesson 11 of 75
Introduction

Roots and coefficients

Objective. Use Vieta-style relationships for higher polynomials.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

For a polynomial equation with roots α, β, γ, ..., the relationships between roots and coefficients generalise Vieta's formulae from quadratics to higher degree. For a cubic ax³+bx²+cx+d=0 with roots α, β, γ, the sum α+β+γ = −b/a, the sum of pairwise products αβ+βγ+γα = c/a, and the product αβγ = −d/a. These identities let expressions like α²+β²+γ² be found without solving the cubic directly, using α²+β²+γ² = (α+β+γ)² − 2(αβ+βγ+γα).

Worked example

The cubic 2x³ − 5x² + 3x − 1 = 0 has roots α, β, γ. Find the value of α + β + γ and αβγ.

2 … 5
Illustrated card showing 2 … 5
  1. 1For ax³+bx²+cx+d=0, sum of roots = −b/a
  2. 2Here a=2, b=−5, so α+β+γ = −(−5)/2 = 5/2
  3. 3Product of roots = −d/a
  4. 4Here d=−1, so αβγ = −(−1)/2 = 1/2

Answer: α+β+γ = 5/2, αβγ = 1/2

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Common mistake. Students forget the alternating signs, writing the sum of products as −c/a instead of c/a. Always derive the signs by expanding a(x−α)(x−β)(x−γ) and matching coefficients rather than guessing.

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