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# Monomial Calculator + Online Solver With Free Steps

The **Monomial Calculator** is a free tool that helps to find the monomial form of the given algebraic expression. The calculator takes the details regarding the expression as input.

**Monomials** are those expressions that have only one term. This one term can be a number, variable, or product of numbers and variables. Any expression having more than one term cannot be a monomial.

The **calculator** returns the monomial expression and can also be used to perform basic operations between monomials.

## What Is a Monomial Calculator?

**A Monomial Calculator is an online calculator that can simplify your algebraic expression by extracting the monomial expression for the given problem.**

The algebraic expressions are commonly used in problems like determining features, modeling buildings, financial analysis, business, sports, and physical motions. These mathematical expressions have deep roots in areas of **engineering**, **business**, and **machine learning**.

Solving such expressions can be quite challenging, therefore it is required to bring these expressions in a simplified form such as **monomial** expression. That is where this **calculator **comes in, it is an efficient tool capable of solving such expressions.

It is a **free** online calculator that you can use multiple times for your problems. This widget doesn’t require any downloading or installation and can be used directly in the browser.

## How To Use the Monomial Calculator?

You can use the **Monomial Calculator **to get the monomial form by putting the target expressions in the respective tabs. The calculator can handle one expression at a time.

One additional **feature** this calculator has is that you can use it to perform various operations between monomial expressions. For example, the addition of two monomial expressions. This further increases the worth of this handy tool.

The calculator has a simple **interface** with one input box and a click-button. You only need to enter the expression in the box and with a single click, you will be presented with the most accurate results.

The calculator is a fairly user-friendly tool that everyone can use. You must follow the detailed instructions to correctly use the **Monomial Calculator** that are written below.

### Step 1

Enter the algebraic expression in the box with the label **“Enter the Equation.” **In case of expression with multiple terms use brackets to differentiate between each term.

### Step 2

Press the **Simplify **button to get the desired solution.

### Output

The output has two sections. The first section is the **input interpretation,** which is what the calculator interpreted about the given expression. It helps users to further confirm the input and clear any ambiguity to avoid errors.

The second section is **results **that display the required monomial expression for the problem. For expressions that cannot be perfectly converted to monomial form, the calculator gives the reduced form by simplifying it as much as possible.

## How Does the Monomial Calculator Work?

This calculator works by **simplifying **the given polynomial expression into a **monomial**. It also simplifies complex monomial expressions. When there is a requirement to solve complicated expressions, this calculator helps to solve those expressions.

Monomial is the type of polynomial expression, so we should know about the polynomial and its types.

### What Is a Polynomial?

A polynomial is an algebraic expression in which the exponents of all the variables are **whole numbers**. The exponents** cannot **be a negative number or a fraction. It consists of variables and constants.

Polynomials are essential in all branches of mathematics, especially in calculus. They can be considered a dialect of mathematics.

#### Terms of a Polynomial

The** terms** of the polynomials are those parts of the expression that** arithmetic **operators separate. However, there are two types of terms that are like terms and unlike terms.

Like terms are those terms that have equal power and the same variable and unlike terms are those that have different power or variables. Polynomials are classified mainly into** three **types based on their terms.

### Monomial

Monomial is defined as the algebraic expression consisting of **one** term that includes constants, variables, or both that are multiplied together. Monomials are the building blocks of polynomials.

Mono means “one,” so these expressions contain only one term. There are three properties of monomials which are given below:

- The power or exponent of variables in a monomial must be a
**positive**integer. - It is essential to have only one
**non-zero**term in the monomial expression. - A monomial can not contain any variable in the
**denominator**.

#### Degree of a Monomial

The degree of a monomial is equal to the** sum** of the exponents of all the variables. It is necessary to be a non-negative integer. For instance, the degree of a monomial given by $abc^2$ is equal to** four**.

The monomial can be linear, quadratic, or cubic based on its degree.

#### Rules of Monomials

When it is a requirement to simplify monomials, the following are **two** rules that should be kept in mind.

- A monomial when multiplied with another monomial, it also results in another monomial expression.
- When a monomial is multiplied by a constant, it also produces another monomial.

#### Multiplying Monomial

Multiplying a Monomial is a method to multiply the monomial with other polynomials. This method follows **distributive law,** in which a monomial is multiplied by each term of other polynomials.

The coefficient is multiplied with the coefficient and the variable is multiplied with the variable. After multiplying, the addition or subtraction of** like** terms takes palace to simplify it further.

When there is a multiplication of monomials with the same variable having their exponents, all of the exponents will be **added** together.

#### Dividing Monomial

Dividing monomials is the process of dividing monomials with other polynomials by **expanding** the terms of both expressions and then canceling out the common terms. The variable is divided by the variable and the same is the case for coefficients.

When the division of monomials with the same base takes place, their exponents will be **subtracted** as per the exponent rules.

### Binomial

A binomial is an algebraic expression that consists of** two **unlike terms having constants and variables. Arithmetic operators join the terms in these expressions.

The coefficients of the terms in the binomial expansion are called **Binomial coefficients**. These are positive integers. The binomial coefficient of the kth term of any binomial expression raised to power $n$ is given by the following formula:

\[^nC_k = \frac {n!}{k!(n-k)!} \]

### Trinomial

An algebraic expression containing** three non-zero **terms and having more than one variable is called Trinomial.

The** perfect square trinomial** is a special expression that is obtained by **squaring** a binomial expression. It is written in standard form as $ax^2+bx+c$.

## Applications of the Monomial

Monomials have vast real-life applications. They are used by career professionals who want to make complex calculations. For instance, an engineer would use polynomials to design the curves for designing a roller coaster.

**Monomials** are also used to describe traffic patterns so that proper traffic plans can be implemented. They are an essential tool for economists to model their economic growth.

Medical researchers apply monomials to relate the behavior of bacterial colonies.

## History

Initially, all the equations involved in the equations are written in the form of **words** instead of variables and numbers. In the 15th century, a mathematical form with variables and coefficients came into existence.

In 1544 for the first time, signs for sum and subtraction were used by **Michael Stifel**. Later in 1557, the notation for equality was also introduced. The polynomial equation was introduced in 1963 by **Rene Descartes**.

These polynomial equations used starting alphabets such as a,b, and c to represent constants and last alphabets like x,y, and z to represent variables. The word polynomial was derived from the Greek word **“poly”** which means many terms.

So using different signs and notation resulted in polynomial expression, which was the sum of many singular terms. These single terms are called **monomials**. Now monomial terms are considered the most simplified form of algebraic expressions.

## Solved Examples

The best way to analyze the working of a calculator is to solve some examples using it. Let’s discuss some examples solved by the **Monomial Calculator**.

### Example 1

A machine learning researcher is working on a regression problem. The model he trained is overfitted for which he has to simply the following expression.

\[ 21 x^2 y^7 \, – \, 9 x^5 y^4 \]

The goal is to determine a monomial expression with a single term.

### Solution

The solution is a simplified expression of the problem.

\[ 3 x^2 y^4 \, (7 y^3 – 3 x^3) \]

### Example 2

Consider the following expression.

\[ (3z^5) . (9z^7) \]

Find the result of this monomial product using the calculator.

### Solution

The result is obtained using simply the power technique. If expressions with the same bases are multiplied, then add the powers.

\[ 27 z^{12} \]

Here, the coefficients with the variables are considered constant and are separately multiplied to find the product.

### Example 3

A college student in his mathematics exam is presented with a trinomial expression given by $2x^3-3x^2+1$. He is asked to simplify it into a monomial expression.

### Solution

The given expression can easily be simplified using a **monomial calculator** by just inserting it in the provided space. The simplified expression is given below:

\[(x-1)^2(2x+1)\]