What is Vieta's Formula
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Last updated on October 4, 2025

Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots. These formulas are named after the French mathematician François Viète. In this topic, we will learn about Vieta's formulas and how they can be applied to polynomials.

Understanding Vieta's Formula

Vieta's formulas provide a powerful tool for relating the coefficients of a polynomial to the sums and products of its roots. Let’s explore how to apply Vieta's formulas to find these relationships.

Vieta's Formula for Quadratic Equations

For a quadratic equation of the form\( ( ax^2 + bx + c = 0 ),\) Vieta's formulas are: - The sum of the roots  \(r_1 \) and \( r_2 \) is given by: \(( r_1 + r_2 = -\frac{b}{a} )\) 

The product of the roots is given by:\( ( r_1 \cdot r_2 = \frac{c}{a} )\)

Vieta's Formula for Cubic Equations

For a cubic equation of the form \(( ax^3 + bx^2 + cx + d = 0 )\), Vieta's formulas are:


The sum of the roots\( ( r_1, r_2, r_3 )\) is: (\(( r_1 + r_2 + r_3 = -\frac{b}{a} ) \)

The sum of the products of the roots taken two at a time is: \(( r_1r_2 + r_2r_3 + r_1r_3 = \frac{c}{a} ) \)

The product of the roots is: \(( r_1 \cdot r_2 \cdot r_3 = -\frac{d}{a} )\)

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Vieta's Formula for Higher-Degree Polynomials

For a polynomial of degree  n :

The sum of the roots taken one at a time is: \(( -\frac{\text{coefficient of } x^{n-1}}{\text{leading coefficient}} )\)

The sum of the products of the roots taken two at a time is: \(( \frac{\text{coefficient of } x^{n-2}}{\text{leading coefficient}} ) \) And so on, until the product of the roots (with alternating signs) is given by the constant term divided by the leading coefficient.

Importance of Vieta's Formulas

Vieta's formulas are crucial in algebra and allow for the analysis and understanding of polynomial roots without explicitly solving the polynomial.

They provide insights into relationships between coefficients and roots, useful in various mathematical and real-world applications.

Tips and Tricks to Memorize Vieta's Formulas

  • Memorizing Vieta's formulas can be simplified by understanding the pattern of relationships between coefficients and roots.
  • Remember that the sum of roots is related to the negative of the second highest coefficient, and products relate directly to the constant term.
  • Practice with different equations to reinforce these concepts.

Common Mistakes and How to Avoid Them While Using Vieta's Formulas

When applying Vieta's formulas, students might encounter several pitfalls. Here are some mistakes and ways to avoid them:

Problem 1

Find the sum and product of the roots of the equation \( 3x^2 - 5x + 2 = 0 \).

Okay, lets begin

The sum of the roots is\( (\frac{5}{3}) \)and the product is \((\frac{2}{3}).\)

Explanation

For the equation \(( 3x^2 - 5x + 2 = 0 ):\) 

The sum of the roots: \(( r_1 + r_2 = -\frac{-5}{3} = \frac{5}{3} )\) 

The product of the roots:\( ( r_1 \cdot r_2 = \frac{2}{3} )\)

Well explained 👍

Problem 2

For the cubic polynomial \( x^3 - 6x^2 + 11x - 6 = 0 \), use Vieta's formulas to find the sum of the roots.

Okay, lets begin

The sum of the roots is 6.

Explanation

The sum of the roots of the cubic polynomial \(( x^3 - 6x^2 + 11x - 6 = 0 ) \) is given by:\( ( r_1 + r_2 + r_3 = -\frac{-6}{1} = 6 )\)

Well explained 👍

Problem 3

Determine the product of the roots for the polynomial ( 2x³- 3x² + x - 5 = 0 ).

Okay, lets begin

The product of the roots is 5/2.

Explanation

The product of the roots for \(( 2x^3 - 3x^2 + x - 5 = 0 ) \) is: \(( r_1 \cdot r_2 \cdot r_3 = -\frac{-5}{2} = \frac{5}{2} )\)

Well explained 👍

Problem 4

Given the quadratic equation 4x² + 8x + 3 = 0 , find the sum and product of the roots.

Okay, lets begin

The sum of the roots is -2 and the product is 3/4.

Explanation

For the equation\( ( 4x^2 + 8x + 3 = 0 ):\) - The sum of the roots: \(( r_1 + r_2 = -\frac{8}{4} = -2 ) \)

The product of the roots: \(( r_1 \cdot r_2 = \frac{3}{4} )\)

Well explained 👍

Problem 5

For the equation ( x³+ 7x² + 14x + 8 = 0 ), use Vieta's formulas to determine the sum of the roots.

Okay, lets begin

The sum of the roots is -7.

Explanation

The sum of the roots for the cubic equation\( x^3 + 7x^2 + 14x + 8 = 0 \) is: \(( r_1 + r_2 + r_3 = -\frac{7}{1} = -7 )\)

Well explained 👍

FAQs on Vieta's Formulas

1.What is Vieta's formula for a quadratic equation?

For a quadratic equation\( ( ax^2 + bx + c = 0 )\), Vieta's formulas are: 

Sum of roots: \\(( r_1 + r_2 = -\frac{b}{a} ) \)

Product of roots: \(( r_1 \cdot r_2 = \frac{c}{a} )\)

2.How do you apply Vieta's formulas to a cubic polynomial?

For a cubic polynomial \(( ax^3 + bx^2 + cx + d = 0 )\), Vieta's formulas are: 

Sum of roots: \(( r_1 + r_2 + r_3 = -\frac{b}{a} ) \)

Sum of products of roots taken two at a time: \(( r_1r_2 + r_2r_3 + r_1r_3 = \frac{c}{a} ) \)

Product of roots: \(( r_1 \cdot r_2 \cdot r_3 = -\frac{d}{a} )\)

3.Are Vieta's formulas applicable to higher-degree polynomials?

Yes, Vieta's formulas can be extended to higher-degree polynomials, where relationships between coefficients and sums/products of roots follow a systematic pattern.

4.What are common mistakes with Vieta's formulas?

Common mistakes include incorrectly identifying coefficients, sign errors, and confusing sums with products. Always rewrite the polynomial in standard form and carefully apply the formulas.

5.Do Vieta's formulas apply to non-polynomial equations?

No, Vieta's formulas apply only to polynomial equations. They are not suitable for rational or transcendental equations.

Glossary for Vieta's Formulas

  • Roots: Values of the variable that satisfy the equation.
  • Coefficient: A numerical factor in terms of a polynomial.
  • Quadratic Equation: A polynomial equation of degree 2.
  • Cubic Equation: A polynomial equation of degree 3.

Jaskaran Singh Saluja

About the Author

Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.

Fun Fact

: He loves to play the quiz with kids through algebra to make kids love it.