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# Rational Expression|Definition & Meaning

## Definition

A **rational expression** is a fraction whose **numerator** and **denominator** are **polynomials.** Just like its name, The **ratio** of two polynomials is named the **rational expression**.

When **f** is an **expression** that is **rational** then it can also be noted in the form, **x****/y** where x and y are **polynomials**. Basic arithmetic operations like **subtraction, addition, multiplication**, and **division** can be done on rational expressions just like polynomials or any other expression.

A good principle of rational **expressions** is that when any of the **above-mentioned** operations are performed on two rational expressions, the result is always a different **rational expression**. Unlike polynomials, rational expressions are normally easy to multiply or divide but relatively harder to add or **subtract.** An example is given below.

## What Is a Rational Expression?

You would have studied **rational** numbers, which are written in the form of p/q. But on the other hand, Rational **expressions** are the ratio of two polynomials.

We have to put the **polynomial** expression equal to zero in order to find the **root or zero** of the expression. But if one desires to figure out the zeros of rational expressions, only the **numerator** should be made zero when the expression is **reduced to its lowest term.**

The lowest terms indicate that numerator and denominator both don’t have even a single **factor in common.** Like when we talk about a fraction, then 4/16 is not in the lowest form of the **expression.** It can be reduced even further by taking 2 as a **common** factor. Finally, 1/4 is the lowest form.

Likewise, for **example,** $\mathsf{\dfrac{x^2+2x}{3x}}$ which is a rational expression, but it is not in its lowest form. So again if we take the common **factor** x from both the **numerator** and **denominator,** we get $\mathsf{\dfrac{x+2}{3}}$, which is now the lowest form of the **expression.**

## Rational Expressions Simplification

Some significant points to memorize for performing arithmetic operations on rational fractions are listed below. These operations include **addition**, **subtraction**, **division,** and **multiplication.**

When getting **rational expressions** to a reduced form, the basic and most important step is to **factor** both the numerator and the denominator. And then, **cancel** out the common factors of the **expressions.**

### Addition and Subtraction

In solving **Rational** expressions, if the fraction has the same polynomial in the **denominator, then **add or subtract the **numerators **while keeping the denominator unchanged. then **if possible**, further reduce the expression.

If the **denominators** of a rational expression are not the same, the first step should be taking the **LCM**. Now, make each rational expression equal by making the denominator of all expressions identical**.** In the final step, Add or subtract the terms**,** then **if possible**, further reduce the expression.

### Multiplication

**Factorize** the polynomials in fractions (Numerator and Denominator), then **reduce** wherever possible. **Multiplying** the remaining **numerators** and denominators separately will result in a **reduced form.**

### Division

Multiply the **inverted** denominator with the first **rational** expression (i.e., numerator) because reduction is much easier after **converting** the division into multiplication by **inverting** the denominator, just like in the case of dividing fractions. As explained above, further **simplification** is identical to **multiplication.**

## Addition and Subtraction of Rational Expressions

As we already know, for adding or subtracting any two fractions, the **denominator** should be the **same** for both fractions. The same rule is functional to rational **functions** too. Normally, the given formula is how we represent addition and subtraction:

\[ \mathsf{\dfrac{a}{c} + \dfrac{b}{c} = \dfrac{a+b}{c}} \]

\[ \mathsf{\dfrac{a}{c} – \dfrac{b}{c} = \dfrac{a-b}{c}} \]

Assuming a fraction. For example, add and subtract 2/3 and 1/2.

So first, adding both fractions, we get:

\[ \mathsf{\dfrac{2}{3} + \dfrac{1}{2} = \dfrac{4+3}{6} = \dfrac{6}{6}} \]

Now by subtracting 1/2 from 2/3, we get:

\[ \mathsf{\dfrac{2}{3} – \dfrac{1}{2} = \dfrac{4-3}{6} = \dfrac{1}{6}} \]

In the above steps, you can see that we take the LCM first to make the numerators identical, and once the expressions are normalized, we perform the operations.

## Multiplication and Division of Rational Expressions

Just like we added and subtracted **rational functions,** we can multiply and divide them too. The general formula is:

\[ \mathsf{\dfrac{a}{b} \times \dfrac{c}{d} = \dfrac{ac}{bd}} \]

\[ \mathsf{\dfrac{a}{b} \div \dfrac{c}{d} = \dfrac{ad}{bc}} \]

## The Rule for Rational Fraction

We need to address the **unspoken** rule of dealing with rational expressions. We know that **division of a number by zero** isn’t permitted. The same practice is followed for rational expressions. With rational expressions, we will always take the values for** X** that won’t give a **division by zero.** We sometimes note these restrictions down, but we will always need them to be memorized.

\[ \mathsf{\dfrac{2}{x-2} \hspace{0.5cm};\hspace{0.5cm} \dfrac{y^2 -1}{y^2+5} \hspace{0.5cm};\hspace{0.5cm} \dfrac{z^4 + 18z + 1}{z^2 – z – 6}} \]

In the first one listed above, we cannot take** x=2**. The second rational expression **can not be zero in the denominator,** so we don’t have to stress about any restrictions. Also, note that the second rational expression’s **numerator will be zero, which** is acceptable if we avoid **division by zero.**

Now, for the third rational expression, we will have to avoid** m = 3 and m = −2.** The final rational **expression** given above can never be zero in the denominator, so we don’t need to have any **limitations.**

## Simplifying Rational Expressions

The **quotient** of 2 polynomial expressions is the rational expression. Now the properties of fractions **will also apply** to rational expressions, just like reducing the expressions by cutting common factors from the numerator and the denominator. To be able to do this, first of all, we have to **factor in the numerator and denominator.** Let’s begin with the rational expression given below:

\[ \mathsf{\dfrac{x^2 +8x + 16}{x^2 + 11x + 28}} \]

**Factorizing** the numerator and denominator and rewriting the expression:

\[ \mathsf{\dfrac{(x + 4)^2}{(x + 4) (x + 7)}} \]

By canceling out the common values, we can solve the expression factor (x+4) as:

\[ \mathsf{\dfrac{x+4}{x+7}}\]

## Example of Solving a Rational Expression

Simplify the given expression:

### Solution

The **common** factor can be canceled because 1 is the result when any **expression** is divided by itself.

\[ \mathsf{\dfrac{(x+3) (x-3)} {(x+3) (x+1)}} \]

\[ \mathsf{\dfrac{x-3} {x+1}} \]

*All images/mathematical drawings were created with GeoGebra.*