Dividing Rational Expressions

Introduction

How do we define rational expressions? Rational expressions are fractions that have a polynomial in the numerator, denominator, or both. This means that both the numerator and denominator are polynomials in the case of a rational expression, for example, ( x ² + 2 x ) /3x is a rational expression. We know that the four operations of mathematics are addition, subtraction, multiplication and division. Can we easily perform division on rationale expressions just as we do them for numbers? Let us find out.

Definition

Rational expressions are fractions that have a polynomial in the numerator, denominator, or both. For example, $\frac{x}{x-2}$ is a rational expression as it has polynomials in both its numerator as well as its denominator. Before we start understanding how to add or subtract rational expressions, it is important to recall some important terms and concepts that are integral to rational expressions. 

Factors are the building blocks of multiplication. In the case of rational expressions as well, factors are those algebraic expressions that completely divide the rational expression. For example, the factor of the rational expression $\frac{x^2-1}{x+1}$ will be x + 1 as it is common to both the numerator as well as the denominator. 

Therefore, a number or an algebraic expression can be said to be a factor of a rational expression, if it is common to both the numerator as well as its denominator. 

Terms are single numbers or variables and numbers connected by multiplication. For example, 6, -5, x³ are all terms.

Expressions are groups of terms connected by addition and subtraction. For example, 4x² – 7 is an expression.

Before we move to understand the division of rational expressions, let us learn how to simplify them.

Simplifying Rational Expressions

Simplification of a rational expression is the process of reducing a rational expression in its lowest terms possible. It is similar to reducing fractions or rational numbers to their lowest form. The following steps are sued for simplification of rational expressions-

1. Factorize both the denominator and numerator of the rational expression. Also, remember to write each expression in standard form. 
2. Reduce the expression by cancelling out common factors in the numerator and denominator
3. Rewrite the remaining factors in the numerator and denominator.

Let us understand the above steps by an example.

Example

Simplify the rational expression $\frac{3x^2- 3x}{3x^3- 6x^2+3x}$

Solution

We are given the rational expression

$\frac{3x^2- 3x}{3x^3- 6x^2+3x}$

In order to simplify this rational expression, first we will look for factors that are common to the numerator & denominator. We have, 3x is a common factor the numerator & denominator. Note that it is clear that x ≠0. Hence we will cancel the common factor to get,

$\frac{3x ( x -1 )}{3x ( x^2-2x+1 )}$

Now, if possible, we will look for other factors that are common to the numerator and denominator. We can see that ( x – 1 ) is also a common factor of the numerator and the denominator as x²-2x+1 can be written as ( x – 1 ) ( x – 1 ) .

Therefore, we now have,

$\frac{( x -1 )}{(x-1) ( x-1 )}$

We now have,

$\frac{1}{( x-1 )}$

Therefore, the rational expression $\frac{3x^2- 3x}{3x^3- 6x^2+3x}$ in its simplified form will be $\frac{1}{( x-1 )}$

Let us now learn how to divide rational expressions. 

How to Divide Rational Expressions?

The division of rational expression is done in the same manner as we do for fractions. This means that for dividing rational expressions we need to find the reciprocal of the rationale expression which is the divisor.  When we divide one rational expression with another, there can be two situations – 

1. Rational Expressions have the same denominator
2. Rational Expressions have different denominators

Let us consider both the cases one by one.

Rational Expressions having the Same Denominators

The following steps should be followed while adding or subtracting rational expressions having the same denominators – 

1. Obtain the numerators of the two given rational expressions and their common denominator
2. Find the reciprocal of the rational expression which is the divisor and change the division sign to that of the multiplication sign. 
3. Factorise both the numerators and the denominators of each rational expression, if possible. Check for any common factors between the numerators and the denominators so as to strike them off. We will then be left with rational expressions in simplified form where the numerators and the denominators do not have anything in common.
4. Write a rational expression whose numerator is the product obtained on multiplying the respective numerators and the denominator is the product obtained on multiplying the respective denominators of the given rational expressions.
5. Simplify the rational expression, if required.

Let us understand the above steps with an example.

Example

Solve the rational expressions $\frac{x}{x-1} ÷ \frac{2-x}{x-1}$

Solution

We are required to find the value of $\frac{x}{x-1} ÷ \frac{2-x}{x-1}$

We can clearly see that both the rational expressions have the same denominator, i.e. x – 1.

Therefore, going by the above stated steps, we will find the reciprocal of the rational expression which is the divisor and change the division sign to that of the multiplication sign.  We will then have,

$( \frac{x}{x-1} ) x (  \frac{2-x}{x-1} ) = \frac{x (   2-x )}{(x-1) ( x-1 )} = \frac{2 x- x^2}{x^2 – 2x+1}$

Hence, $\frac{x}{x-1} ÷ \frac{2-x}{x-1} = \frac{2 x- x^2}{x^2 – 2x+1}$

Now, let us see how we will divide rational expressions have different denominators.

Rational Expressions having Different Denominators

In order to divide two rational expressions with different denominators, the following steps will be followed – 

1. Obtain the rational expressions and their denominators. The denominators of the fractions should be such that they are not the same. 
2. Find the reciprocal of the rational expression which is the divisor and change the division sign to that of the multiplication sign. 
3. Factorise both the numerators and the denominators of each rational expression, if possible. Check for any common factors between the numerators and the denominators so as to strike them off. We will then be left with rational expressions in the simplified form where the numerators and the denominators do not have anything in common.
4. Write a rational expression whose numerator is the product obtained on multiplying the respective numerators and the denominator is the product obtained on multiplying the respective denominators of the given rational expressions.
5.  Reduce the resultant rational expression if possible.

Let us understand the above steps with an example.

Example Find the value of 

[ ( x ² + 3 x – 28 ) /  ( x ² + 4 x + 4 ) ] ÷ [ ( x ² – 49 ) /   ( x ² –  5 x – 14 ) ]

Solution We are required to find the value of 

[ ( x ² + 3 x – 28 ) /  ( x ² + 4 x + 4 ) ] ÷ [ ( x ² – 49 ) /   ( x ² –  5 x – 14 ) ]

Let us perform the division of these rational expressions using the above steps. We can see that the given rational expressions are of different denominators. We will first find the reciprocal of the rational expression which is the divisor and change the division sign to that of the multiplication sign.  In this question, the divisor is [ ( x ² – 49 ) /   ( x ² –  5 x – 14 ) ]. Hence we need to find its reciprocal. We know that a reciprocal of a rational expression is found by interchanging the numerator and the denominator. Therefore, the reciprocal of [ ( x ² – 49 ) /   ( x ² –  5 x – 14 ) ] will be 

[ ( x ² –  5 x – 14 ) ( x ² – 49 ) ]

So, [ ( x ² + 3 x – 28 ) /  ( x ² + 4 x + 4 ) ] ÷ [ ( x ² – 49 ) /   ( x ² –  5 x – 14 ) ] ..(1 )

can now be written as  – 

[ ( x ² + 3 x – 28 ) /  ( x ² + 4 x + 4 ) ]  *  [ ( x ² –  5 x – 14 ) ( x ² – 49 ) ]

We will now factorise the numerators and the denominators of each fraction.

( x ² + 3 x – 28 ) =  ( x  –   4 )   ( x + 7 )  ………………… ( 2 )

( x ² + 4 x + 4 ) =  ( x  + 2 )  ( x + 2 )  ……………………. ( 3 )

( x ² – 49 ) =  x ² – 7 ² =  ( x – 7 )  ( x + 7 )   ……………….. ( 4 )

( x ² – 5 x – 14 ) =  ( x – 7 )  ( x  + 2 ) ……………………….. ( 5 )

Substituting the values of ( 2 ) , ( 3 ) , ( 4 )  and ( 5 )  in ( 1 ), we get

  [ ( x – 4 )  ( x + 7 ) /  ( x + 2 )  ( x + 2 ) ]  ÷  [ ( x -7 ) ( x + 7 ) /  ( x – 7 )  ( x + 2 ) ]

Now, we will substitute the divisor with its reciprocal. We will get,

[ ( x – 4 ) ( x + 7 ) /  ( x + 2 ) ( x + 2 ) ] * [ ( x – 7 ) ( x + 2 ) / ( x – 7 ) ( x + 7 ) ]

We can see that there are common terms in the expression which can be cancelled. We will have,

$\frac{( x – 4 ) ( x + 7 )}{ ( x + 2 ) ( x + 2 )} x \frac{( x – 7 ) ( x + 2 )}{ ( x – 7 ) ( x + 7 )}$

= $\frac{x-4}{x+2}$

 Hence,

$\frac{( x^2 + 3 x – 28 )}{ ( x^2 + 4 x + 4 )} ÷ \frac{( x^2– 49 )}{ ( x^2–  5 x – 14 )} = \frac{x-4}{x+2}$

Solved Examples

Example 1 Find the value of $\frac{( x + 2 )}{4 y} ÷ \frac{( x^2  – x – 6 )}{12 y^2}$

Solution We have been given the expression $\frac{( x + 2 )}{4 y} ÷ \frac{( x^2  – x – 6 )}{12 y^2}$

Let us perform the division of these rational expressions using the above steps. We can see that the given rational expressions are of different denominators. We will first find the reciprocal of the rational expression which is the divisor and change the division sign to that of the multiplication sign.  In this question, the divisor is $\frac{( x^2  – x – 6 )}{12 y^2}$. Hence we need to find its reciprocal. We know that a reciprocal of a rational expression is found by interchanging the numerator and the denominator. Therefore, the reciprocal of $\frac{( x^2  – x – 6 )}{12 y^2}$ will be $\frac{12 y^2}{( x^2 – x – 6 )}$

We now have,

$\frac{( x + 2 )}{4 y} ÷ \frac{( x^2  – x – 6 )}{12 y^2} =  \frac{( x + 2 )}{4 y} x \frac{12 y^2}{( x^2  – x – 6 )}$

We will now factorise the numerators and the denominators of each fraction.

( x ² – x – 6 ) = ( x – 3 ) ( x + 2 )

Substituting this value in the original expression we get,

$\frac{( x + 2 )}{4 y} x \frac{12 y^2}{( x^2  – x – 6 )}$

=  $\frac{( x + 2 )}{4 y} x \frac{12 y^2}{( x–3 )(x+2)}$

We can see that there are common terms in the expression which can be cancelled. We will have,

$\frac{( x + 2 )}{4 y} x \frac{3 x 4 y^2}{( x–3 )(x+2)}$

= $\frac{3y}{x-3}$

Hence, $\frac{( x + 2 )}{4y} ÷ \frac{( x^2 – x – 6 )}{12y^2} = \frac{3y}{x-3}$

Example 2 Find the value of 

$\frac{(2 t^2+ 5 t + 3 )}{( 2 t^2 + 7 t + 6 )} ÷ \frac{( t^2 + 6 t + 5 )}{(- 5 t^2  – 35 t – 50 )}$

Solution We need to find the value of 

$\frac{(2 t^2+ 5 t + 3 )}{( 2 t^2 + 7 t + 6 )} ÷ \frac{( t^2 + 6 t + 5 )}{(- 5 t^2  – 35 t – 50 )}$

Let us perform the division of these rational expressions using the above steps. We can see that the given rational expressions are of different denominators. We will first find the reciprocal of the rational expression which is the divisor and change the division sign to that of the multiplication sign.  In this question, the divisor is  $\frac{( t^2 + 6 t + 5 )}{(- 5 t^2  – 35 t – 50 )}$. Hence we need to find its reciprocal. We know that a reciprocal of a rational expression is found by interchanging the numerator and the denominator. Therefore, the reciprocal of $\frac{( t^2 + 6 t + 5 )}{(- 5 t^2  – 35 t – 50 )}$ will be  $\frac{(- 5 t^2  – 35 t – 50 )}{( t^2 + 6 t + 5 )}$. 

We now, have,

$\frac{(2 t^2+ 5 t + 3 )}{( 2 t^2 + 7 t + 6 )} ÷ \frac{( t^2 + 6 t + 5 )}{(- 5 t^2  – 35 t – 50 )}$

= $\frac{(2 t^2+ 5 t + 3 )}{( 2 t^2 + 7 t + 6 )} x \frac{(- 5 t^2  – 35 t – 50 )}{( t^2 + 6 t + 5 )}$

We will now factorise the numerators and the denominators of each fraction.

2 t ² + 5 t + 3 = ( t + 1 ) ( 2 t + 3 )

2 t ² + 7 t + 6 = ( 2 t + 3 ) ( t + 2 )

t ² + 6 t + 5 = ( t + 1 )  ( t + 5 )

-5 t ² – 35 t -50 = -5 ( t ² + 7 t + 10 ) = -5 ( t + 2 ) ( t + 5 )

Substituting this value in the original expression we get,

$\frac{( t + 1) ( 2t + 3 ) }{( 2t + 3 ) ( t + 2 )} x \frac{( t + 1) ( t + 5 )}{ -5 (  t + 2 ) ( t + 5 )}$

We can see that there are common terms in the expression which can be cancelled. We will have,

$\frac{( t + 1) ( 2t + 3 ) }{( 2t + 3 ) ( t + 2 )} x \frac{-5 (  t + 2 ) ( t + 5 )}{ ( t + 1) ( t + 5 )}$

= -5

Hence, 

$\frac{(2 t^2+ 5 t + 3 )}{ ( 2 t^2 + 7 t + 6 )} ÷ \frac{( t^2 + 6 t + 5 )}{ (- 5 t^2  – 35 t – 50 )}$ = – 5

Key Facts and Summary

1. Rational expressions are fractions that have a polynomial in the numerator, denominator, or both.
2. A number or an algebraic expression can be said to be a factor of a rational expression if it is common to both the numerator as well as its denominator.
3. Terms are single numbers or variables and numbers connected by multiplication. 6, -5, x³ are all terms.
4. Expressions are groups of terms connected by addition and subtraction. 4x² – 7 is an expression.