Relationship Between Zeros and Coefficients

A polynomial is an expression built from variables and coefficients using addition, subtraction, and multiplication. For a quadratic polynomial ax² + bx + c, the zeros (also called roots) are the values of x that make the polynomial equal to zero. Understanding how these zeros relate to the coefficients a, b, and c reveals deep algebraic structure without even solving the equation. Vieta's formulas describe the relationship between the zeros of a polynomial and its coefficients. For a quadratic with zeros α and β, the sum α + β equals −b/a, and the product αβ equals c/a. These results are not mere coincidences — they follow directly from the factored form of the quadratic. Knowing them allows you to construct polynomials with desired roots, verify solutions, and solve many competition-style problems efficiently. These relationships extend beyond quadratics to higher-degree polynomials, but mastering the quadratic case is essential. In this article you will derive the formulas from first principles, apply them in worked examples, and see why François Viète's sixteenth-century insight still sits at the heart of algebra today.

Key takeaways
  • Start with the standard quadratic
  • Write the factored form
  • Expand the factored form
  • Match coefficients
  • Reconstruct the polynomial from its roots
  • Important special cases

Start with the standard quadratic

Consider the general quadratic polynomial p(x) = ax² + bx + c, where a ≠ 0 and a, b, c are real numbers. Suppose α and β are its two zeros, meaning p(α) = 0 and p(β) = 0.

Write the factored form

Because α and β are roots, the Factor Theorem tells us that (x − α) and (x − β) are both factors of p(x). Therefore: p(x) = a(x − α)(x − β). The leading coefficient a is kept outside so that the leading term remains ax².

Expand the factored form

Expand the product: a(x − α)(x − β) = a[x² − (α + β)x + αβ] = ax² − a(α + β)x + aαβ. This is now in standard form ax² + bx + c.

Match coefficients

Comparing ax² − a(α + β)x + aαβ with ax² + bx + c term by term: the coefficient of x gives b = −a(α + β), so α + β = −b/a. The constant term gives c = aαβ, so αβ = c/a. These are Vieta's formulas for a quadratic.

Reconstruct the polynomial from its roots

Conversely, if you know two numbers α and β that should be roots of a monic quadratic (a = 1), you can immediately write: x² − (α + β)x + αβ = 0. This gives a quick way to build polynomials with specified zeros without working backwards through the quadratic formula.

Important special cases

If b = 0, the sum of roots is zero, meaning the roots are negatives of each other (e.g., x² − 4 has roots +2 and −2). If c = 0, the product of roots is zero, meaning at least one root is zero (e.g., x² + 3x = x(x + 3) has roots 0 and −3). These observations let you read properties of the roots directly from the coefficients.

The Key Equation

α+β=−ba,αβ=ca\alpha + \beta = -\dfrac{b}{a}, \qquad \alpha\beta = \dfrac{c}{a}
  • a — Leading coefficient (dimensionless): The coefficient of x²; must be nonzero for the polynomial to be quadratic.
  • b — Middle coefficient (dimensionless): The coefficient of x; determines, together with a, the sum of the roots.
  • c — Constant term (dimensionless): The coefficient with no x attached; together with a it gives the product of the roots.
  • \alpha,\,\beta — Zeros (roots) (dimensionless): The two values of x for which the polynomial equals zero.

Worked Example

Problem. The zeros of the quadratic 2x² − 7x + 3 are α and β. Without finding the individual roots, determine: (i) α + β, (ii) αβ, and (iii) α² + β². Then verify by actually solving the equation.

α+β=72,αβ=32,α2+β2=374\alpha+\beta=\tfrac{7}{2},\quad \alpha\beta=\tfrac{3}{2},\quad \alpha^2+\beta^2=\tfrac{37}{4}

Answer. Sum of roots = 7/2, product of roots = 3/2, and α² + β² = 37/4.

Figure
Diagram
xyαβ(α+β)/2midpoint=−b/(2a)p(x)=ax²+bx+cSum: α+β = −b/aProduct: αβ = c/a|←——— α+β ———→|
Origins · 1 minFrançois Viète (Vieta) and how this was discovered
FV
François Viète (Vieta)
1540–1603 · French

Viète was the first mathematician to introduce systematic algebraic notation using letters for both known and unknown quantities, a revolution that made general algebraic reasoning possible. In his 1591 work In Artem Analyticem Isagoge he pioneered this symbolic approach, and his later work De Emendatione Aequationum established what are now called Vieta's formulas — the relationships between the roots of a polynomial and its coefficients. His formulas hold for polynomials of every degree: for degree n, the sum of roots is −a_{n-1}/a_n, and the product of all roots is (−1)^n · a_0/a_n. Viète worked as a lawyer and royal privy councillor for King Henry III and Henry IV of France. During a war with Spain, he used his exceptional mathematical skill to decode encrypted Spanish military messages — essentially serving as a royal cryptanalyst. The Spanish were so baffled that they complained to the Pope, claiming France must be using black magic.

  • The Quadratic Formula — The quadratic formula gives the explicit roots α and β; Vieta's formulas let you verify their sum and product without computing each root individually.
  • Factor Theorem — The derivation of Vieta's formulas depends on the Factor Theorem: if α is a zero of p(x), then (x − α) is a factor, enabling the factored form used in the proof.
  • Discriminant of a Quadratic — The discriminant b² − 4ac can be expressed using Vieta's formulas as a²(α − β)², connecting the nature of the roots to their sum and product.
  • Symmetric Functions of Roots — Expressions like α² + β² and α³ + β³ are symmetric functions that can always be rewritten in terms of α + β and αβ using algebraic identities, directly extending Vieta's formulas.