JUMP TO TOPIC

# Simplest Form (Algebra)|Definition & Meaning

## Definition

The **simplest form** of an **algebraic expression** is when it cannot be simplified further. Therefore, in its **simplest form,** the expression is **straightforward** and easy to use. There are many ways to simplify** algebraic expressions.** For example, 3x + 6x + 9y can instead be written as 9(x + y), which is its **simplest form.**

## What Is Simplest Form?

Anything that has been reduced to the **minimum** possible level or that requires not much effort to reduce it further is known as the **simplest** form. In **mathematical** **algebra**, the **simplest** form is the least attainable fraction of a number or a linear equation.

If we talk about fractions, the **numerator** and the **denominator** have to be simplified up till the last bit in order to say that we have achieved the **simplest** form.

For instance, we have a **fraction**, 1/2, which is in its **simplest** possible formation and can not be further reduced as a **whole** **number** **f****raction**.

## What Are Algebraic Expressions?

The concept of **algebraic** **expressions** is the use of words or **alphanumeric** **characters** to present data without providing their accurate value. We were taught how to represent an unspecified number using **symbols** like x, y, and z in the fundamentals of **mathematics**. Here, we refer to such characters as **variables**.

A calculation may have both constants and variables. **Coefficients** are these number that comes in front of a **variable** and is transformed by it.

**Algebraic** **expressions** are also composed of arithmetic operations such as **addition**, **subtraction**, etc.

An expression consists of three elements, namely, **coefficients**, **variables**, and **constants**. The elements separated by any arithmetic symbol are known as a term. So in the above expression,

- x represents a
**variable**whose actual value is not known and can contain any virtual value. - The number 5 next to the variable x is known as a
**coefficient**. Coefficients are usually constant values and are used with**variables**such as the above. They enhance the value of a variable by increasing it or decreasing it if they are in the denominator. - The last element that is usually separate from the other terms is a
**constant**, and as the name suggests, it is a**well-defined**numerical value. In this case, 3 is a**constant**.

The above expression is known as a **binomial** **expression** which means it has two (bi) elements with no similarity between them.

### Type of Algebraic Expressions

The three most common types of algebraic **expressions** are explained below.

#### Monomial Algebraic Expressions

These are the most basic **expressions** consisting of only one element; that’s why they are known as **monomial** **expressions**. Such as 5y^{2}, 10x, etc.

#### Binomial Algebraic Expressions

These contain two **elements**, both can be variables or can contain one **variable** and one **constant**. Such as 10xy + 3, 5x^{2} – 5y, etc.

#### Polynomial Algebraic Expressions

An expression containing more than two elements is known as a **polynomial** **expression**. All the terms in this expression are different from one another. Such as 2x + 7y + 4z + 6, etc.

## What Is Simplifying?

A straightforward meaning of **simplifying** is to make things easy. In mathematical terms, simplifying is downsizing the **equation**, **expressions**, **fractions**, or any problem to the least **minimum** **form**. The benefit of simplifying **expressions** can help in calculations and solving a particular problem.

Most mathematical **expressions** cause a nightmare for the students, so the idea of simplifying is to lower their pain and guide the subject as efficiently as it can; therefore, simplifying algebraic **expressions** can assist us in making mathematical **calculations** easier.

The steps to follow for simplifying any **expression** are quite easy to comprehend; you will start by removing the symbols of similar groups or **parentheses** and **brackets** in simple terms and then combining the similar terms having the same exponent or maybe the same **variables**. It is an instrumental process of converting a complex form into a **simpler** one.

## Steps for Finding the Simplest Form

The steps for finding the **simplest** form are as very simple as the name suggests.

### For an Algebraic Fraction

- The first step is to find
**factors**that are similar in numerical values or are coefficients of similar variables. Also, look for exponents, and combine the monomials having similar**exponents**. - The next step is to find a fraction in which one of the factors is a
**prime****factor**. Now a prime number is such a number that is divisible only by itself, such that it only generates two**factors**, one and itself. - The last step is to
**multiply**and**divide**by the same number so that its**fraction**is 1. Simply multiplying the fraction with a number will obviously make it larger instead of reducing it, so it is recommended to**multiply**and**divide**by the same**number,**such as 2/2, 5/5, etc.

### For an Algebraic Expression

#### Merge Similar Terms

Identifying **similar** **terms** is pretty straightforward; you just have to look for similar variables or **variables** having the same **exponents, **such as:

** = 10x + 5x**

** = 15x**

If the terms are not similar to any of the available **terms**, then that means the expression is already simplified to the least **minimum, **such as:

** = 6x + 2x ^{2}**

The above expression does have the same variable used, but the **powers** are different, which makes it unlike terms, and thus they cannot be merged into a more **simplified** **form**.

#### Use of Distributive Property

When there are parentheses or **square** **brackets** in an expression, the best approach is to eliminate them first before moving forward. Such a simple **expression** like:

** = 4(5x + 9) + 3x**

can be simplified if the **parenthesis** is removed. In this scenario, the **distributive** **property** can help us to remove the parenthesis by multiplying the number outside with the elements inside the parenthesis. Thus resulting in a long but **simplified** **expression**.

Let’s simplify the above expression by applying the **property**,

## Solved Example of a Simple form of Algebraic Expression

Simplify the expression given below

\[ \dfrac{(x^2 – x)(x^2 + 5x + 6)}{(2x + 6)(x^2 – 1)} \]

### Solution

The first step is to figure out the factors of **the numerator** and **denominator**. The numerator can be factorized as:

**x(x – 1) . (x + 3)(x + 2)**

Similarly, the denominator can be written after factorizing as:

**2(x + 3) . (x + 1)(x – 1)**

Now simply eliminating similar terms from the fraction gives us the following:

\[ \dfrac{(x\cancel{(x – 1)}).\cancel{(x + 3)}(x + 2)}{(2\cancel{(x + 3)}).\cancel{(x – 1)}(x + 1)} \]

Now we can multiply the remaining terms of the fraction to get:

\[ \dfrac{x(x + 2)}{2(x + 1)} \]

\[ = \dfrac{x^2 + 2x}{2(x + 1)} \]

Usually, when we simplify **fractions**, we leave the **denominator** as it is, whereas the **numerator** gets multiplied and forms a **whole** **expression**. This practice can be helpful when you encounter a situation in which you require the least common **multiple** of a **complex** equation and have **denominators** with similar factors.

*All images are created using GeoGebra. *