Rational Expression

Rational Terms

We have studied that the fraction is made up of two parts numerator and denominator. When the numerator and denominator consists of polynomials such fractions are called as rational expressions.

For example

  • \[\frac{4x + 2}{3x + 3}\]

  • \[\frac{5x + 7}{x + 2}\]

 We can also perform arithmetic operations such as addition, subtraction and multiplication with the rational terms. Just like the fraction, the rationals can be reduced to the simplified lowest rational terms. These polynomial equations could have more than one degree.

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In this article let us discuss polynomials and rational expressions and simplifying rational expressions.

What is Rational Expressions?

Rational expression can also be said as the ratio of two polynomials expressions.

If p (x) and q (x) are two polynomials, with q (x) ≠ 0, then general form of rational expression is 

p (x) / q (x) 


In rational expression both the numerator and denominator are polynomials.The denominator of rational expression can never be zero. Like algebraic expressions it contains unknown variables. To simplify the rational expression we reduce it to its simplest form. 

 Simplifying Rational Expressions

The quotient of two polynomial expressions is called a rational expression. 

Simplifying the rational expression means to reduce to its lowest terms. Rational expression is in its lowest form if all common factors from the numerator and denominator have been canceled. 

1. We first need to factor the polynomials

2. Cancel any common factors from the numerator and denominator of the rational expression

For Example :

3x2 + x

__________

4x2 + x

Factor the numerator and denominator 

    x( 3x + 1)

= ________

    x( 4x + 1)

Now, simplify the expression by canceling the common factor   

    (3x + 1)

=   ______

    (4x + 1)

It is the lowest form.

Multiplying Rational Expressions

Multiplication of rational expressions works the same way as multiplication in fractions. We multiply the numerators with the numerator and denominator with the denominator to find the product. Before multiplying, factor the numerators and denominators just as we did when simplifying rational expressions it will make the calculations easier. We can also simplify the product of rational expressions.

Steps to multiply two rational expressions

  1. Factor the numerator and denominator

  2. Multiply numerators.

  3. Multiply denominators.

  4. Simplify

For example :

[\[\frac{(x + 1)}{2x}\]] X [\[\frac{(y + 2)}{(x + 5)}\]]

= \[\frac{(x + 1) X (y + 2)}{2x X (x + 5)}\]

= \[\frac{xy + 2x + y + 2}{2x2 + 10x}\]

Dividing Rational Expressions

Division of rational expressions works the same way as division in fractions. To divide a rational expression by another rational expression, we have to multiply the first expression by the reciprocal of the second expression. 

Steps to Divide Two Rational Expression

  1. Write the first rational expression multiplied by the reciprocal of the second.

  2. Factor the numerators and denominators.

  3. Multiply the numerators.

  4. Multiply the denominators.

  5. Simplify.

For Example

(2x2 + x - 6/x2 - 1) (x2 - 4 / x2 + 2x -1)

Solution:

Write the first rational expression multiplied by the reciprocal of the second.

(2x2 + x - 6/x2 - 1)  x ( x2 + 2x - 1/ x2 - 4)

Factor the numerators and denominators.

[(2x -3)( x + 2)/ (x + 1)( x- 1) ] x [ (x+1)2/(x + 2)( x - 2)]

Cancel the common factors

[(2x + 3)(x + 1) / ( x- 1)(x -2)]

Adding and Subtracting Rational Expressions

Adding and subtracting rational expressions works just like adding and subtracting general fractions. To add fractions, we need to find a common denominator.

We have to rewrite the fractions with a common denominator before we are able to add. We must do the same thing when subtracting rational expressions.

The general rule for adding or subtracting the rational expression is 

\[\frac{a}{b}\] + \[\frac{c}{d}\] = \[\frac{a \times d + b \times c}{b \times d}\]

The easiest common denominator to use will be the least common denominator, or LCD. The LCD is the least common multiple that the denominators have in common.

Steps to Add or Subtract the Rational Expressions are

  1. Factor the numerator and denominator.

  2. Find the LCD of the expressions.

  3. Multiply the expressions by LCD that changes the denominators to the LCD.

  4. Add or subtract the numerators.

  5. Simplify.

For example:

Add 5/x + 2/y

Solution: first let us find the LCD i.e xy

Now taking xy as the denominator and dividing it by the individual denominator and multiplying by each fraction.

(5/x) x (y/y) + (2/y) x (x/x)

= 5y/xy + 2x/xy

= (5y + 2x) / xy

Solved Examples

  1. Multiply ((x + 1)2 / 8x2 ) x ( 4x / (x + 1))

Solution: 

=[(x + 1)( x + 1) / 4 x 2 x]  x [ 4 x / (x + 1)]

Cancel the common factors

=( x + 1) / 2x

  1. Divide 10 / (3x + 1) (2 x + 7) 8 / (3x + 1) (x + 2)

Solution:

Take the reciprocal of second rational expression and multiply

=[2 / ( 3x + 1)( 2x + 7)]  x [ ( 3x + 1) ( x+ 2) /8)]

 cancel the common factors

= 5 (x + 2) / 4( 2x + 7)

Quiz Time

Solve the following

  1. 5x2 / 7x 10x2 / 14x

  2. [(x - 1)(x2 + 2xy + y2) / 3x + 2 ] x [ (x + 1)/ (x + 1)2

FAQ (Frequently Asked Questions)

1. How do you Solve a Rational Expression?

Answer: Rational expressions contain fractions with polynomials in both the numerator and the denominator. Solving rational expression equations is more complex than solving standard polynomial equations because you have to find the common denominator of the rational terms, then simplify the resulting expressions. Cross-multiplication transforms these equations into regular polynomial equations.

Steps to add or subtract the rational expressions are

  1. Factor the numerator and denominator.

  2. Find the LCD of the expressions.

  3. Multiply the expressions by LCD that changes the denominators to the LCD.

  4. Add or subtract the numerators.

  5. Simplify.