Polynomial Roots Calculator (2024)

  • Calculators
  • ::
  • Polynomial Calculators
  • ::
  • Polynomial Roots Calculator

Polynomial Roots Calculator (1)Polynomial Roots Calculator (2)

This free math tool finds the roots (zeros) of a given polynomial.The calculator computes exact solutions for quadratic, cubic, and quartic equations.Calculator shows all the work and provides step-by-step on how to find zeros and their multiplicities.

working...

Polynomial Calculators

  • FactoringPolynomials

  • Polynomial Roots
  • Synthetic Division
  • PolynomialOperations
  • GraphingPolynomials
  • Simplify Polynomials
  • Generate From Roots

Rational Expressions

  • Simplify Expression

  • Multiplication / Division
  • Addition / Subtraction

Radical Expressions

  • Rationalize Denominator

  • Simplifying

Solving Equations

  • Quadratic Equations Solver

  • Polynomial Equations
  • Solving Equations - WithSteps

Quadratic Equation

  • Solving (with steps)

  • Quadratic Plotter
  • Factoring Trinomials

2D Shapes

  • Equilateral Triangle

  • Right Triangle
  • Oblique Triangle
  • Square Calculator
  • Rectangle Calculator
  • Circle Calculator
  • Hexagon Calculator
  • Rhombus Calculator
  • Trapezoid Calculator

3D Shapes

  • Cube

  • Cuboid
  • Triangular Prism
  • Pyramid
  • Cylinder
  • Cone
  • Sphere

Analytic Geometry

  • Distance calculator

  • Midpoint Calculator
  • Triangle Calculator
  • Graphing Lines
  • Lines Intersection
  • Two Point Form
  • Line-Point Distance
  • Parallel/Perpendicular
  • Circle Equation
  • Ellipse

  • Circle From 3 Points
  • Circle-line Intersection

Complex Numbers

  • Modulus, inverse, polar form

  • Division
  • SimplifyExpression

Systems of equations

  • System 2x2

  • System 3x3
  • System 4x4

Matrices

  • Add, Subtract,Multiply
  • Determinant Calculator
  • Matrix Inverse
  • CharacteristicPolynomial
  • Eigenvalues
  • Eigenvectors
  • MatrixDecomposition

Calculus Calculators

  • Limit Calculator

  • Derivative Calculator
  • Integral Calculator

Sequences & Series

  • ArithmeticSequences

  • GeometricSequences
  • Find nth Term

Trigonometry

  • Degrees toRadians

  • Trig.Equations

Numbers

  • Long Division

  • Evaluate Expressions
  • Fraction Calculator
  • Greatest Common Divisor GCD
  • Least Common Multiple LCM
  • Prime Factorization
  • Scientific Notation
  • Percentage Calculator
  • Dec / Bin / Hex

Statistics and probability

  • Probability Calculator
  • Probability Distributions
  • Descriptive Statistics

  • Standard Deviation
  • Z - score Calculator
  • NormalDistribution
  • T-Test Calculator

Financial Calculators

  • Simple Interest

  • Compound Interest
  • AmortizationCalculator
  • Annuity Calculator

Other Calculators

  • Sets

  • Work Problems

Hire MATHPORTAL experts to do math homework for you.Prices start at $3 per problem.

EXAMPLES

example 1:ex 1:

find roots of the polynomial $4x^2 - 10x + 4$

example 2:ex 2:

find polynomial roots $-2x^4 - x^3 + 189$

example 3:ex 3:

solve equation $6x^3 - 25x^2 + 2x + 8 = 0$

example 4:ex 4:

find polynomial roots $2x^3-x^2-x-3$

example 5:ex 5:

find roots $2x^5-x^4-14x^3-6x^2+24x+40$

Find more worked-out examples in the database of solved problems..

Search our database with more than 250 calculators

TUTORIAL

How to find polynomial roots ?

The process of finding polynomial roots depends on its degree. The degree is the largest exponent in the polynomial. For example,the degree of polynomial $ p(x) = 8x^{\color{red}{2}} + 3x -1 $ is $\color{red}{2}$.

We name polynomials according to their degree. For us, the most interesting ones are:quadratic - degree 2, Cubic - degree 3, and Quartic - degree 4.

Roots of quadratic polynomial

This is the standard form of a quadratic equation

$$ a\,x^2 + b\,x + c = 0 $$

The formula for the roots is

$$ x_1, x_2 = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a} $$

Example 01: Solve the equation $ 2x^2 + 3x - 14 = 0 $

In this case we have $ a = 2, b = 3 , c = -14 $, so the roots are:

$$ \begin{aligned}x_1, x_2 &= \dfrac{-b \pm \sqrt{b^2-4ac}}{2a} \\x_1, x_2 &= \dfrac{-3 \pm \sqrt{3^2-4 \cdot 2 \cdot (-14)}}{2\cdot2} \\x_1, x_2 &= \dfrac{-3 \pm \sqrt{9 + 4 \cdot 2 \cdot 14}}{4} \\x_1, x_2 &= \dfrac{-3 \pm \sqrt{121}}{4} \\x_1, x_2 &= \dfrac{-3 \pm 11}{4} \\x_1 &= \dfrac{-3 + 11}{4} = \dfrac{8}{4} = 2 \\x_2 &= \dfrac{-3 - 11}{4} = \dfrac{-14}{4} = -\dfrac{7}{2}\end{aligned}$$

solve using calculator

Quadratic equation - special cases

Sometimes, it is much easier not to use a formula for finding the roots of a quadratic equation.

Example 02: Solve the equation $ 2x^2 + 3x = 0 $

Because our equation now only has two terms, we can apply factoring.Using factoring we can reduce an original equation to two simple equations.

$$ \begin{aligned}2x^2 + 3x &= 0 \\\color{red}{x} \cdot \left( \color{blue}{2x + 3} \right) &= 0 \\\color{red}{x = 0} \,\,\, \color{blue}{2x + 3} & \color{blue}{= 0} \\\color{blue}{2x } & \color{blue}{= -3} \\\color{blue}{x} &\color{blue}{= -\frac{3}{2}}\end{aligned}$$

solve using calculator

Example 03: Solve equation $ 2x^2 - 10 = 0 $

This is also a quadratic equation that can be solved without using a quadratic formula.

.$$ \begin{aligned}2x^2 - 18 &= 0 \\2x^2 &= 18 \\x^2 &= 9 \\\end{aligned}$$

The last equation actually has two solutions. The first one is obvious

$$ \color{blue}{x_1 = \sqrt{9} = 3} $$

and the second one is

$$ \color{blue}{x_2 = -\sqrt{9} = -3 }$$

Roots of cubic polynomial

To solve a cubic equation, the best strategy is to guess one of three roots.

Example 04: Solve the equation $ 2x^3 - 4x^2 - 3x + 6 = 0 $.

Step 1: Guess one root.

The good candidates for solutions are factors of the last coefficient in the equation. In this example, the last number is -6 so our guesses are

1, 2, 3, 6, -1, -2, -3 and -6

if we plug in $ \color{blue}{x = 2} $ into the equation we get,

$$ 2 \cdot \color{blue}{2}^3 - 4 \cdot \color{blue}{2}^2 - 3 \cdot \color{blue}{2} + 6 = \\\\ 2 \cdot 8 - 4 \cdot 4 - 6 - 6 = 0$$

So, $ \color{blue}{x = 2} $ is the root of the equation. Now we have to divide polynomial with $ \color{red}{x - \text{ROOT}} $

In this case we divide $ 2x^3 - x^2 - 3x - 6 $ by $ \color{red}{x - 2}$.

$$ ( 2x^3 - 4x^2 - 3x + 6 ) \div (x - 2) = 2x^2 - 3 $$

Now we use $ 2x^2 - 3 $ to find remaining roots

$$ \begin{aligned}2x^2 - 3 &= 0 \\2x^2 &= 3 \\x^2 &= \frac{3}{2} \\x_1 & = \sqrt{ \frac{3}{2} } = \frac{\sqrt{6}}{2}\\x_2 & = -\sqrt{ \frac{3}{2} } = - \frac{\sqrt{6}}{2}\end{aligned}$$

solve using calculator

Cubic polynomial - factoring method

To solve cubic equations, we usually use the factoting method:

Example 05: Solve equation $ 2x^3 - 4x^2 - 3x + 6 = 0 $.

Notice that a cubic polynomial has four terms, and the most common factoring method for such polynomials is factoring by grouping.

$$ \begin{aligned}2x^3 - 4x^2 - 3x + 6 &=\color{blue}{2x^3-4x^2} \color{red}{-3x + 6} = \\&= \color{blue}{2x^2(x-2)} \color{red}{-3(x-2)} = \\&= (x-2)(2x^2 - 3)\end{aligned}$$

Now we can split our equation into two, which are much easier to solve.The first one is $ x - 2 = 0 $ with a solution $ x = 2 $, and the second one is$ 2x^2 - 3 = 0 $.

$$\begin{aligned}2x^2 - 3 &= 0 \\x^2 = \frac{3}{2} \\x_1x_2 = \pm \sqrt{\frac{3}{2}}\end{aligned}$$

RESOURCES

1. Roots of Polynomials — find roots of linear, quadratic and cubic polynomials.

2. Rational Root Theorem with examples and explanations.

439 507 212 solved problems

Polynomial Roots Calculator (2024)

References

Top Articles
Latest Posts
Article information

Author: Kerri Lueilwitz

Last Updated:

Views: 6392

Rating: 4.7 / 5 (47 voted)

Reviews: 94% of readers found this page helpful

Author information

Name: Kerri Lueilwitz

Birthday: 1992-10-31

Address: Suite 878 3699 Chantelle Roads, Colebury, NC 68599

Phone: +6111989609516

Job: Chief Farming Manager

Hobby: Mycology, Stone skipping, Dowsing, Whittling, Taxidermy, Sand art, Roller skating

Introduction: My name is Kerri Lueilwitz, I am a courageous, gentle, quaint, thankful, outstanding, brave, vast person who loves writing and wants to share my knowledge and understanding with you.