Topic 1 of 8 · Form 5

Roots of polynomial equations

For ax² + bx + c = 0 with roots α, β: α + β = −b/a; αβ = c/a.

Notes for this topic 1 file, in order

For ax² + bx + c = 0 with roots α, β: α + β = −b/a; αβ = c/a.

For ax³ + bx² + cx + d = 0 with roots α, β, γ: Σα = −b/a; Σαβ = c/a; αβγ = −d/a.

For quartics: Σα = −b/a, Σαβ = c/a, Σαβγ = −d/a, αβγδ = e/a.

α² + β² + γ² = (Σα)² − 2Σαβ; 1/α + 1/β + 1/γ = Σαβ/αβγ; α³ + β³ + γ³ via Sₙ recurrences.

Worked example: The roots of x³ − 2x² + 3x − 4 = 0 are α, β, γ. Find α² + β² + γ².

Σα = 2; Σαβ = 3 → α² + β² + γ² = 4 − 6 = −2 (so not all roots are real)

To find an equation with roots 2α, 2β, 2γ, substitute x = y/2; for roots α + 1 etc., substitute x = y − 1; for roots α², use x = √y and rearrange.

Worked example: Find the equation whose roots are 2α, 2β, 2γ for the cubic above.

(y/2)³ − 2(y/2)² + 3(y/2) − 4 = 0 → y³ − 4y² + 12y − 32 = 0

Exam tip: A negative value of α² + β² + γ² for a real cubic shows that some roots are complex.

Show answers

No questions about this topic yet.