# Polynomial Calculator

## Calculate polynomials step by step

The calculator will find (with steps shown) the sum, difference, product, and result of the division of two polynomials (quadratic, binomial, trinomial, etc.). It will also calculate the roots of the polynomials and factor them. Both univariate and multivariate polynomials are accepted.

First polynomial:

Second polynomial:

The second polynomial is needed for addition, subtraction, multiplication, division; but not for root finding, factoring.

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please write it in the comments below.

The Polynomial Calculator is a versatile tool for performing various polynomial operations quickly and accurately. Polynomials are very important in algebra, and understanding them is crucial to mastering mathematics.

## How to Use the Polynomial Calculator?

• ### Input

Enter one or two polynomials. The second polynomial is needed for binary operations like addition and subtraction.

• ### Calculation

Once you've entered the polynomial(s), click the "Calculate" button.

• ### Result

The calculator will quickly display the result in the output section. This could be a factored polynomial,  the roots of the polynomial(s), the result of addition, subtraction, multiplication, and division.

## What Is Meant by Polynomial?

A polynomial is a mathematical expression consisting of the sum of terms. A term is the product of a constant coefficient and a non-negative integer power of a variable. It can be expressed in the general form as follows:

$$P(x)=a_nx^n+a_{n-1}x^{n-1}+\ldots+a_2x^2+a_1x+a_0,$$

where:

• $a_n,a_{n-1},\ldots,a_2,a_1,a_0$ are the constant coefficients.
• $x$ is the variable.
• $n$ is a non-negative integer known as the degree of the polynomial.

## Breakdown of the Elements of a Polynomial

In a polynomial, each element has a specific role and characteristic. Let's dive into the individual parts that make up the polynomial:

• Terms

These are individual units in a polynomial. Consider the polynomial $3x^2-4x+7$. Here, the terms are $3x^2$, $-4x$, and $7$.

• Coefficients

These numbers precede the variable within a term. From our example, $3$, $-4$, and $7$ are the coefficients.

• Variables

These are often represented by letters that stand in for unknown values. In many polynomials, $x$ is a frequently used variable.

• Degree

This refers to the highest power of the variable in the polynomial. For instance, the degree of the polynomial $2x^3-5x^2+x-8$ is $3$.

## Polynomial Classification by the Number of Terms

• Monomial: A polynomial with just one term. Example: $7x^5$.
• Binomial: A polynomial with two terms. Example: $x^3-4x$.
• Trinomial: A polynomial with three terms. Example: $3x^2-x+9$.

## Operations on Polynomials

With polynomials, you can perform various arithmetic operations, just like with numbers. The basic operations are addition, subtraction, and multiplication. Let's understand each of them with the help of formulas and examples:

When adding polynomials, you combine like terms. "Like terms" are terms with the same variables raised to the same powers:

$$\left(a_nx^n+a_{n-1}x^{n-1}+\ldots+a_2x^2+a_1x+a_0\right)+\left(b_nx^n+b_{n-1}x^{n-1}+\ldots+b_2x^2+b_1x+b_0\right)=\left(a_n+b_n\right)x^n+\left(a_{n-1}+b_{n-1}\right)x^{n-1}+\ldots+\left(a_2+b_2\right)x^2+\left(a_1+b_1\right)x+\left(a_0+b_0\right)$$

Example:

$$(2x^2+3x-5)+(x^2-x+4)=(2+1)x^2+(3-1)x+(-5+4)=3x^2+2x-1$$
• Subtraction of Polynomials

For subtraction, you subtract the like terms:

$$\left(a_nx^n+a_{n-1}x^{n-1}+\ldots+a_2x^2+a_1x+a_0\right)-\left(b_nx^n+b_{n-1}x^{n-1}+\ldots+b_2x^2+b_1x+b_0\right)=\left(a_n-b_n\right)x^n+\left(a_{n-1}-b_{n-1}\right)x^{n-1}+\ldots+\left(a_2-b_2\right)x^2+\left(a_1-b_1\right)x+\left(a_0-b_0\right)$$

Example:

$$(3x^2+5x-8)-(x^2+2x-3)=(3-1)x^2+(5-2)x+\left(-8-(-3)\right)=2x^2+3x-5$$
• Multiplication of Polynomials

When multiplying polynomials, multiply each term in the first polynomial by each term in the second polynomial and add the resulting terms. For two binomials, $a+b$ and $c+d$, the product is

$$(a+b)\cdot(c+d)=ac+ad+bc+bd$$

Example:

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

Understanding these operations is crucial to solving algebraic problems. They lay the foundation for advanced algebraic concepts.

## Why Choose Our Polynomial Calculator?

• ### Accuracy

Our calculator uses advanced algorithms to ensure you always get accurate results.

• ### User-Friendly Interface

Designed with simplicity in mind, the platform offers an intuitive interface that even those new to polynomials can easily navigate.

• ### Versatility

Beyond just basic polynomial calculations like factoring and root-finding, our tool supports a variety of operations, including addition, subtraction, multiplication, and division.

• ### Speed

Our calculator provides instant solutions, saving you time and effort.

### FAQ

#### What is the Polynomial Calculator used for?

The Polynomial Calculator is designed to perform a variety of polynomial operations, including addition, subtraction, multiplication, and division, providing accurate and instant solutions to your polynomial problems.

#### What is a polynomial?

A polynomial is a mathematical expression involving a sum of non-negative integer powers in one or more variables multiplied by some coefficients.

#### What is zero of a polynomial?

A polynomial's zero (or root) is the value of the polynomial's variable for which the polynomial evaluates to zero. In other words, if $P(c)=0$, then $c$​ is a zero of the polynomial $P(x)$.

#### What is the Fundamental Theorem of Algebra?

The fundamental theorem of algebra states that every non-constant single-variable polynomial equation with complex coefficients has at least one complex root. A polynomial of degree $n$ will have exactly $n$ roots, counting multiplicity.