Vieta's Formulas Analyzer

Math Algebra • Polynomial and Rational Functions

Written by STEM Calculators Team Published December 14, 2025 Updated February 24, 2026

Enter a polynomial and press Calculate.

Extracts Vieta identities from coefficients.
Computes approximate roots (real/complex) to verify the identities numerically.
Plots \(y=p(x)\) and labels intercepts as coordinates.

Drag to pan. Mouse wheel zooms (cursor-centered). Double-click resets view. Intercepts are labeled as \((x,0)\) and \((0,p(0))\) when enabled.

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8. Vieta's Formulas Analyzer — Theory

Vieta’s formulas connect polynomial coefficients to symmetric combinations of its roots (sum, pairwise products, product, etc.). They are essential for quick root-relation problems and for verifying factorizations.

1) Setup

Let \[ p(x)=a_n x^n + a_{n-1}x^{n-1}+\cdots+a_1 x + a_0,\qquad a_n\neq 0, \] and suppose its (possibly complex) roots are \(r_1,r_2,\dots,r_n\) (counted with multiplicity). Then \[ p(x)=a_n\,(x-r_1)(x-r_2)\cdots(x-r_n). \]

2) Vieta’s formulas (elementary symmetric sums)

Expanding the factored form and matching coefficients gives:

\[ \begin{aligned} r_1+r_2+\cdots+r_n &= -\frac{a_{n-1}}{a_n},\\[4pt] \sum_{i

More compactly, the \(k\)-th elementary symmetric sum \(e_k\) satisfies \[ e_k(r_1,\dots,r_n)=(-1)^k\,\frac{a_{n-k}}{a_n}\qquad (k=1,2,\dots,n). \]

3) Monic case

If the polynomial is monic (\(a_n=1\)), the formulas become especially simple: \[ r_1+\cdots+r_n=-a_{n-1},\quad \sum_{i

4) Quick example

For \(p(x)=x^3-6x^2+11x-6\) (monic cubic), \[ r_1+r_2+r_3=6,\qquad r_1r_2+r_1r_3+r_2r_3=11,\qquad r_1r_2r_3=6. \] Indeed the roots are \(1,2,3\), so \(1+2+3=6\), \(1\cdot2+1\cdot3+2\cdot3=11\), and \(1\cdot2\cdot3=6\).

5) Intercepts and what they mean

The y-intercept is \((0,p(0))=(0,a_0)\). Real roots \(r\) give x-intercepts \((r,0)\). Complex roots do not appear as x-intercepts on a real graph, but they still participate in Vieta relations.

6) Notes and limits

Vieta relations are exact from coefficients, even if roots are irrational or complex.
Numerical root-finding can introduce tiny rounding errors; small imaginary parts may appear for real roots.
Repeated roots are counted with multiplicity in all sums/products.

Frequently Asked Questions

What are Vieta's formulas for a polynomial?

Vieta's formulas relate the coefficients of p(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_0 to symmetric sums of the roots r1, ..., rn. For example, r1 + ... + rn = -a_{n-1}/a_n and r1 r2 ... rn = (-1)^n a_0/a_n.

How does this calculator find Vieta relations from my polynomial?

It reads the coefficients of your expanded polynomial and matches them to the standard Vieta identities for elementary symmetric sums. The relations are exact from coefficients even when roots are irrational or complex.

Why do I see tiny imaginary parts for roots that should be real?

Numerical root-finding can introduce small rounding errors, so a real root may appear with a very small imaginary component. The tolerance setting helps decide when a root should be treated as real.

What do the intercept labels on the graph represent?

Real roots r correspond to x-intercepts (r, 0) on the plot of y = p(x). The y-intercept is (0, p(0)) which equals (0, a_0) for an expanded polynomial.

Do Vieta's formulas work with repeated or complex roots?

Yes, the identities include all roots counted with multiplicity, and complex roots are included in the sums and products. Complex roots do not appear as x-intercepts on a real graph, but they still satisfy the Vieta relations.