Sums and Products of Roots
Master Vieta's Formulas — the powerful shortcut to find sums, products, and symmetric functions of polynomial roots without solving for each root individually.
1. What Are Roots of a Polynomial?
A root (also called a zero) of a polynomial f(x) is any value of x for which f(x) = 0. Graphically, roots are the x-intercepts of the polynomial curve.
By the Fundamental Theorem of Algebra, every polynomial of degree n has exactly n roots in the complex number system, counting repeated roots with multiplicity.
- Root / Zero: a value of x where f(x) = 0
- Repeated Root: a root with multiplicity greater than 1
- Real Roots: roots on the real number line
- Complex Roots: roots of the form a + bi
Example: For f(x) = x² − 5x + 6 = (x − 2)(x − 3), the roots are 2 and 3.
2. Introducing Vieta's Formulas
Vieta's Formulas connect the coefficients of a polynomial directly to sums and products of its roots, without solving the polynomial explicitly.
💡 Core Idea
If a polynomial is written as f(x) = aₙ(x − r₁)(x − r₂)...(x − rₙ), then expanding and comparing coefficients gives identities for sums and products of roots.
These identities are associated with François Viète and later generalized in broader algebraic work.
3. Quadratic Equations
Consider the quadratic equation:
Let the roots be α and β. Then:
SUM OF ROOTS
Negative second coefficient over leading coefficient
PRODUCT OF ROOTS
Constant term over leading coefficient
Special Case: Monic Quadratic (a = 1)
For x² + bx + c = 0:
- Sum of roots: α + β = −b
- Product of roots: αβ = c
This gives the standard monic form: x² − (α+β)x + αβ = 0
🔍 Show Full Derivation
Step 1: Expand the factored form:
Step 2: Compare with ax² + bx + c:
- −a(α+β) = b → α+β = −b/a
- aαβ = c → αβ = c/a
4. Cubic Equations
For a cubic with roots α, β, γ:
Vieta's Formulas for Cubics
For degree n, the product of all roots is (−1)ⁿ × a₀ / aₙ.
5. General Polynomial (Degree n)
For a degree-n polynomial with roots r₁, r₂, ..., rₙ:
| Symmetric Sum | Notation | Formula |
|---|---|---|
| Sum of all roots | r₁ + r₂ + ... + rₙ | −aₙ₋₁ / aₙ |
| Sum of products of pairs | Σ(rᵢrⱼ), i<j | aₙ₋₂ / aₙ |
| Sum of triple products | Σ(rᵢrⱼrₖ), i<j<k | −aₙ₋₃ / aₙ |
| Product of all roots | r₁r₂...rₙ | (−1)ⁿ × a₀ / aₙ |
The k-th elementary symmetric sum equals (−1)ᵏ × aₙ₋ₖ / aₙ.
6. Worked Examples
Question: Without solving, find the sum and product of roots of 2x² − 7x + 3 = 0.
Here, a = 2, b = −7, c = 3.
Sum of roots = −b/a = 7/2
Product of roots = c/a = 3/2
Question: Form a quadratic whose roots are 3 + √5 and 3 − √5.
Sum = 6
Product = 4
Equation: x² − 6x + 4 = 0
Question: Given x² − 5x + 6 = 0, find α² + β².
α + β = 5, αβ = 6
α² + β² = (α + β)² − 2αβ = 25 − 12 = 13
Question: The roots of x³ − 6x² + kx − 8 = 0 are α, β, γ and αβγ = 8. Find k.
By Vieta: α + β + γ = 6 and αβγ = 8.
Taking roots 2, 2, 2 gives the correct sum and product.
So αβ + βγ + αγ = 4 + 4 + 4 = 12, hence k = 12.
Question: If α, β are roots of 3x² − x − 2 = 0, find an equation with roots 1/α and 1/β.
α + β = 1/3 and αβ = −2/3
New sum = (α + β)/(αβ) = −1/2
New product = 1/(αβ) = −3/2
New equation: 2x² + x − 3 = 0
7. Forming Equations with New Roots
A major use of Vieta's Formulas is building new equations from transformed roots without explicitly solving the original equation.
| Original roots | New roots | New Sum | New Product |
|---|---|---|---|
| α, β | α + k, β + k | (α+β) + 2k | (α+k)(β+k) |
| α, β | kα, kβ | k(α+β) | k²αβ |
| α, β | 1/α, 1/β | (α+β)/αβ | 1/αβ |
| α, β | α², β² | (α+β)² − 2αβ | (αβ)² |
| α, β | −α, −β | −(α+β) | αβ |
- Let the new variable describe the new root relation.
- Express the old root in terms of the new variable.
- Substitute into the original equation and simplify.
8. Quick Reference Cheat Sheet
α + β = −b/a
αβ = c/a
α+β+γ = −b/a
αβ+βγ+αγ = c/a
αβγ = −d/a
α²+β² = (α+β)²−2αβ
α³+β³ = (α+β)³−3αβ(α+β)
(α−β)² = (α+β)²−4αβ
x² − Sx + P = 0
S = sum of roots, P = product
9. Practice Quiz
Test your understanding of sums and products of roots.
For the equation 3x² + 12x − 9 = 0, what is the sum of the roots?
For 5x² − 3x + 10 = 0, what is the product of the roots?
Which equation has roots 4 and −7?
If α and β are roots of x² − 4x + 1 = 0, find α² + β².
For 2x³ − x² + 3x − 5 = 0, find αβγ.
If α, β are roots of x² − 6x + 4 = 0, what is the product of the new roots (α − 1) and (β − 1)?