Building a quadratic from its zeroes

Now go backwards: you know the sum S and product P of the zeroes, and you want the quadratic.

Since x² – (α + β)x + αβ has zeroes α and β, the answer is x² – Sx + P. Watch the minus in front of the sum.

• Sum –3, product 2: x² – (–3)x + 2 = x² + 3x + 2.
• Sum 1/4, product –1: x² – ¼x – 1. Multiply through by 4 to clear the fraction: 4x² – x – 4.

Multiplying by 4 doesn't change the zeroes. In fact k(x² – Sx + P) works for any non-zero k, so there are infinitely many quadratics with the same zeroes. Choose k to clear the fractions.

Cubics have three relations. If α, β, γ ('gamma') are the zeroes of ax³ + bx² + cx + d:

Check with 2x³ – 5x² – 14x + 8, whose zeroes are 4, –2 and ½:

• sum 4 – 2 + ½ = 5/2 = –(–5)/2 ✓
• pairs –8 – 1 + 2 = –7 = –14/2 ✓
• product 4 × (–2) × ½ = –4 = –8/2 ✓

Notice the signs go minus, plus, minus.