Form a third degree polynomial function with real coefficients calculator

Cubic Equation Calculator

A cubic equation has the form ax3 + bx2 + cx + d = 0. It is defined as third degree polynomial equation. It must have the term in x3 or it would not be cubic but any or all of b, c and d can be zero. The solutions of this cubic equation are termed as the roots or zeros of the cubic equation. All third degree polynomial equations will have either one or three real roots. Solve the roots of the third degree equation using this cubic equation calculator.

A cubic equation has the form ax3 + bx2 + cx + d = 0. It is defined as third degree polynomial equation. It must have the term in x3 or it would not be cubic but any or all of b, c and d can be zero. The solutions of this cubic equation are termed as the roots or zeros of the cubic equation. All third degree polynomial equations will have either one or three real roots. Solve the roots of the third degree equation using this cubic equation calculator.

Code to add this calci to your website

Cubic Equation Formula:

x1=(- term1 + r13*cos(q3/3) ) x2=(- term1 + r13*cos(q3+(2*Π)/3) ) x3=(- term1 + r13*cos(q3+(4*Π)/3) ) Where, discriminant(Δ) = q3 + r2 term1 = √(3.0)*((-t + s)/2) r13= 2 * √(q) q = (3c- b2)/9 r = -27d + b(9c-2b2) s = r + √(discriminant) t = r - √(discriminant)

This calculator will help you dynamically to calculate the roots of the cubic equation.

• 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.
It also displays the step-by-step solution with a detailed explanation.

Enter polynomial:

= 0

Examples:

x^2 - 4x + 3

2x^2 - 3x + 1

x^3 – 2x^2 – x + 2

EXAMPLES

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

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

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

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

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

Search our database of more than 200 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:

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

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.

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

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

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

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 $.

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

227 986 418 solved problems

What is a polynomial function with real coefficients?

What is a third degree polynomial function?