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Sums and Products of Roots of Polynomials — Definition, Formula & Examples

Sums and products of roots of polynomials are relationships, known as Vieta's formulas, that express the sum, product, and other symmetric combinations of a polynomial's roots directly in terms of its coefficients — without needing to find the roots themselves.

For a polynomial anxn+an−1xn−1+⋯+a1x+a0=0a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 = 0 with roots r1,r2,…,rnr_1, r_2, \ldots, r_n, Vieta's formulas state that the elementary symmetric polynomials of the roots equal signed ratios of coefficients: ∑iri=−an−1an\sum_{i} r_i = -\frac{a_{n-1}}{a_n}, ∑i<jrirj=an−2an\sum_{i<j} r_i r_j = \frac{a_{n-2}}{a_n}, and in general, ∑1≤i1<⋯<ik≤nri1⋯rik=(−1)kan−kan\sum_{1 \le i_1 < \cdots < i_k \le n} r_{i_1} \cdots r_{i_k} = (-1)^k \frac{a_{n-k}}{a_n}, with the product of all roots being r1r2⋯rn=(−1)na0anr_1 r_2 \cdots r_n = (-1)^n \frac{a_0}{a_n}.

Key Formula

r1+r2=−ba,r1⋅r2=car_1 + r_2 = -\frac{b}{a}, \qquad r_1 \cdot r_2 = \frac{c}{a}
Where:
  • aa = Leading coefficient of the quadratic $ax^2 + bx + c$
  • bb = Coefficient of the linear term
  • cc = Constant term
  • r1,r2r_1, r_2 = The two roots of the quadratic equation

How It Works

For a quadratic ax2+bx+c=0ax^2 + bx + c = 0 with roots r1r_1 and r2r_2, Vieta's formulas give r1+r2=−bar_1 + r_2 = -\frac{b}{a} and r1⋅r2=car_1 \cdot r_2 = \frac{c}{a}. For a cubic ax3+bx2+cx+d=0ax^3 + bx^2 + cx + d = 0 with roots r1,r2,r3r_1, r_2, r_3, you get r1+r2+r3=−bar_1 + r_2 + r_3 = -\frac{b}{a}, r1r2+r1r3+r2r3=car_1 r_2 + r_1 r_3 + r_2 r_3 = \frac{c}{a}, and r1r2r3=−dar_1 r_2 r_3 = -\frac{d}{a}. The pattern extends to any degree: the kk-th symmetric sum of the roots uses the coefficient an−ka_{n-k} divided by the leading coefficient ana_n, with an alternating sign (−1)k(-1)^k. These formulas let you answer questions about roots using only the coefficients.

Worked Example

Problem: For the polynomial 2x3−10x2+14x−6=02x^3 - 10x^2 + 14x - 6 = 0, find the sum of the roots, the sum of the products of roots taken two at a time, and the product of all three roots.
Identify coefficients: Here a=2a = 2, b=−10b = -10, c=14c = 14, d=−6d = -6.
2x3−10x2+14x−62x^3 - 10x^2 + 14x - 6
Sum of roots: Apply the formula for the sum of roots of a cubic.
r1+r2+r3=−ba=−−102=5r_1 + r_2 + r_3 = -\frac{b}{a} = -\frac{-10}{2} = 5
Sum of pairwise products: Use the second Vieta relation for a cubic.
r1r2+r1r3+r2r3=ca=142=7r_1 r_2 + r_1 r_3 + r_2 r_3 = \frac{c}{a} = \frac{14}{2} = 7
Product of all roots: Use the third Vieta relation.
r1r2r3=−da=−−62=3r_1 r_2 r_3 = -\frac{d}{a} = -\frac{-6}{2} = 3
Answer: The sum of the roots is 55, the sum of pairwise products is 77, and the product of all three roots is 33.

Why It Matters

Vieta's formulas appear throughout competition math and standardized tests whenever a problem asks about roots without requiring you to solve the equation. They are also foundational in precalculus and abstract algebra, where symmetric functions of roots connect polynomial factoring to field theory.

Common Mistakes

Mistake: Forgetting the alternating sign and writing the sum of roots as ba\frac{b}{a} instead of −ba-\frac{b}{a}.
Correction: Each symmetric sum carries a factor of (−1)k(-1)^k. For the sum of roots (k=1k=1), the sign is negative: −an−1an-\frac{a_{n-1}}{a_n}.