Polynomials • Topic 4 of 4

Roots of Polynomials

You rarely need to find the roots themselves — CAT wants symmetric functions of them, and Vieta’s relations deliver those straight from the coefficients. For a quadratic ax² + bx + c with roots α, β: α + β = −b/a and αβ = c/a. For a cubic ax³ + bx² + cx + d with roots α, β, γ: α + β + γ = −b/a, αβ + βγ + γα = c/a, and αβγ = −d/a. From these you build everything else: α² + β² + γ² = (Σα)² − 2Σαβ, and 1/α + 1/β + 1/γ = Σαβ / αβγ. A high-value identity for roots is α³ + β³ + γ³ − 3αβγ = (α+β+γ)(α²+β²+γ²−αβ−βγ−γα), which collapses to 3αβγ whenever the roots sum to zero. The exam trap is sign errors in −b/a and −d/a, so always divide through by the leading coefficient first and track signs carefully.

✅ Solved examples

1. The roots of x² − 7x + 10 = 0 are α and β. Find α² + β².
α + β = 7, αβ = 10. α² + β² = (α+β)² − 2αβ = 49 − 20 = 29.
2. For x³ − 6x² + 11x − 6 = 0, find the sum and product of the roots.
Sum = −(−6)/1 = 6; product = −(−6)/1 = 6.
3. If α, β, γ are roots of x³ − 2x² + 3x − 4 = 0, find 1/α + 1/β + 1/γ.
Σαβ = 3, αβγ = 4. 1/α + 1/β + 1/γ = Σαβ/αβγ = 3/4.
4. If α, β, γ are roots of x³ + x − 1 = 0, find α³ + β³ + γ³.
Σα = 0 (no x² term), so α³+β³+γ³ = 3αβγ. Here αβγ = −(−1)/1 = 1, so the sum = 3·1 = 3.

✏️ Practice — try these, take hints as needed

1. Roots of x² − 5x + 6 = 0 are α, β. Find α² + β².
α + β = 5, αβ = 6.
(α+β)² − 2αβ.
25 − 12.
13
2. For 2x² − 8x + 6 = 0, find the sum and product of roots.
Divide concept: sum = −b/a, product = c/a.
Sum = 8/2.
Product = 6/2.
Sum = 4, product = 3
3. For x³ − 3x² + 4x − 2 = 0, find α + β + γ and αβγ.
Σα = −b/a.
αβγ = −d/a.
Mind the signs of b and d.
Σα = 3, αβγ = 2
4. If α, β, γ are roots of x³ − 4x² + 5x − 2 = 0, find 1/α + 1/β + 1/γ.
Σαβ = c/a = 5.
αβγ = −d/a = 2.
Ratio Σαβ/αβγ.
5/2
5. If the roots of x³ + 6x = 9 sum to S and have product P, find S and P.
Rewrite as x³ + 0x² + 6x − 9 = 0.
No x² term ⇒ Σα = 0.
αβγ = −(−9)/1.
S = 0, P = 9

📝 Topic test — 8 questions

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