Polynomial Calculator is a free online tool that displays the addition, subtraction, multiplication, and division of two polynomials. BYJU’S online polynomial calculator tool makes the calculation faster, and it displays the resultant polynomial in a fraction of seconds for various operations such as addition, subtraction, multiplication, division, derivative and integration.
How to Use the Polynomial Calculator?
The procedure to use the
polynomial calculator is as follows:
Step 1: Enter the polynomials in the respective input field and select required operator
Step 2: Now click the button “Calculate ” or “Multiply” or “Divide” to get the polynomial
Step 3: Finally, the resultant polynomial will be displayed in the new window
What is Meant by Polynomial?
In Mathematics, a polynomial is defined as an algebraic expression that consists of different terms, variables, coefficients, constants and non-negative integer exponents. Also, different arithmetic operations can be performed on polynomials such as addition, subtraction, multiplication and division. A polynomial can be of different types, namely
- Monomial – An expression with a single term. Example: 6x
- Binomial – An expression with two terms separated by mathematical operations. Example: 3x+4y
- Trinomial – An expression with three terms. Example: 2x + 3y – 6z
We can also derive the derivative and antiderivative of the polynomial functions.
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- -4x^3+6x^2+2x=0
- 6+11x+6x^2+x^3=0
- 2x^5+x^4-2x-1=0
- 11+6x+x^2=-\frac{6}{x}
- x^3-2x=0
- 2x^5+x^4-2x-1=0
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Degree of the function:
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( The degree is the highest power of an x. )
Symmetries:
axis symmetric to the y-axis
point symmetric to the origin
y-axis intercept
Roots / Maxima / Minima /Inflection points:
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Characteristic points:
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Slope at given x-coordinates:
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- Calculators
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- 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 $.
$$ \begin{aligned} 2x^2 - 3 &= 0 \\ x^2 = \frac{3}{2} \\ x_1x_2 = \pm \sqrt{\frac{3}{2}} \end{aligned} $$
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