- Calculators
- ::
- Polynomial Calculators
- ::
- Polynomial Roots Calculator
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
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