Roots of Polynomials

Relationships between roots and coefficients

For a polynomial, there are neat relationships (Vieta's formulas) between its roots and its coefficients, so you can find sums and products of roots without solving the equation.

Quadratic: ax² + bx + c = 0

With roots α and β:

α + β = −b/a       (sum of roots)
αβ = c/a           (product of roots)

Cubic: ax³ + bx² + cx + d = 0

With roots α, β, γ:

Σα = α + β + γ = −b/a
Σαβ = αβ + βγ + γα = c/a
αβγ = −d/a

Quartic: ax⁴ + bx³ + cx² + dx + e = 0

With roots α, β, γ, δ:

Σα = −b/a,   Σαβ = c/a,   Σαβγ = −d/a,   αβγδ = e/a

Note the alternating signs (−, +, −, +).

Useful derived results

You can find symmetric functions of the roots from these sums/products, e.g.:

α² + β² = (α + β)² − 2αβ
1/α + 1/β = (α + β)/(αβ)

Forming a new equation

To find an equation whose roots are a transformation of the originals (e.g. 2α, 2β), substitute: if the new root is y = 2x, then x = y/2 — substitute x = y/2 into the original equation and simplify. This is a common exam technique.

Worked example

The quadratic x² − 5x + 6 = 0 has roots α and β. Find α² + β² without solving.

• Sum α + β = 5; product αβ = 6.
• α² + β² = (α + β)² − 2αβ = 5² − 2(6) = 25 − 12 = 13. ✓

Common mistakes

• Getting the signs wrong (sum of roots for a quadratic is −b/a).
• Forgetting the alternating sign pattern for cubics/quartics.
• Trying to solve the polynomial when the question wants a symmetric function of the roots.

Exam tips

• Learn the sum/product relationships for quadratics, cubics and quartics.
• Rewrite symmetric expressions (like α² + β²) in terms of Σα and Σαβ.
• To transform roots, substitute for x in terms of the new variable.

Key facts to remember

• Quadratic: α + β = −b/a, αβ = c/a.
• Cubic: Σα = −b/a, Σαβ = c/a, αβγ = −d/a (signs alternate for higher degrees).
• Find symmetric functions via identities like α² + β² = (α+β)² − 2αβ; transform roots by substitution.