Roots and coefficients — building an equation from its roots

A recurring reverse-direction task: given the roots (or a passing point), find the original quadratic or cubic's constants.

The two directions, and which method fits

FROM ROOTS TO EQUATION: if r₁ and r₂ are roots of a quadratic, the quadratic is a(x − r₁)(x − r₂) = 0 — expand and match to ax² + bx + c to find a, b, c.
FROM A CUBIC'S ROOTS: the same idea extends — a cubic with roots r₁, r₂, r₃ is a(x − r₁)(x − r₂)(x − r₃) = 0, expanded and matched term by term.
FROM GIVEN POINTS: substitute each (x, y) pair into the general form to get one equation per point, then solve the resulting equations simultaneously for the unknown constants.
SUM AND PRODUCT OF ROOTS: for ax² + bx + c = 0, the sum of roots = −b/a and the product of roots = c/a — useful for finding an expression like a² + b² without solving for the roots individually.

Worked example Finding a cubic from three given roots A cubic 3x³ + ax² + bx + c has roots −2, 1/3, and 4. Find a, b, and c. Write the cubic as 3(x + 2)(x − 1/3)(x − 4) = 0, using the leading coefficient 3 already given. Expand (x + 2)(x − 4) = x² − 2x − 8 first, then multiply by (x − 1/3): x³ − 2x² − 8x − (1/3)x² + (2/3)x + 8/3 = x³ − (7/3)x² − (22/3)x + 8/3. Multiply the whole expression by 3 (the leading coefficient): 3x³ − 7x² − 22x + 8. Match to 3x³ + ax² + bx + c: a = −7, b = −22, c = 8. ✓ Expand in stages (pair two factors first) rather than all three at once — it is far easier to check for sign errors one step at a time. ✕

Forgetting the leading coefficient when building from roots

Writing a quadratic as just (x − r₁)(x − r₂) = 0 only works when the leading coefficient is 1. If the question gives a leading coefficient other than 1 (like the "3" in 3x³ + ax² + bx + c), it must be multiplied through AFTER expanding the root factors, or every constant found will be wrong by that factor.

What you are given	Method to use
Two roots of a quadratic	a(x − r₁)(x − r₂) = 0, expand and match
Three roots of a cubic	a(x − r₁)(x − r₂)(x − r₃) = 0, expand in stages
Two points the curve passes through	Substitute each point into the general form, solve simultaneously
An expression like a² + b² using roots a, b	Use sum of roots = −b/a and product = c/a, then (a+b)² − 2ab

Practice for this standard covers the whole topic — flashcards and quiz.