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# Dividing Rational Expressions â€“ Techniques & Examples

Rational expressions in mathematics can be defined as fractions in which either or both the numerator and the denominator are polynomials.Â Just like dividing fractions, **rational expressions are divided by applying the same rules and procedures.**

To divide two fractions, we multiply the first fraction by the inverse of the second fraction. This is done by changing from the division sign (Ã·) to the multiplication sign (Ã—).

* The general formula for dividing fractions and rational expressions is;*

- a/b Ã· c/d = a/b Ã— d/c = ad/bc

For example;

- 5/7 Ã· 9/49 = 5/7 Ã— 49/9

= (5 Ã— 49)/ (7 Ã— 9) = 245/63

= 35/9

- 9/16 Ã· 5/8

= 9/16 Ã— 8/5

= (9 Ã— 8)/ (16 Ã— 5)

= 72/80

= 9/10

## How to Divide Rational Expressions?

Dividing rational expressions follow the same rule of dividing two numerical fractions.

*The steps involved in dividing two rational expressions are:*

- Factor both the numerators and denominators of each fraction. You must know how to factor quadratic and cubic equations.
- Change from division to multiplication sign and flip the rational expressions after the operation sign.
- Simplify the fractions by canceling common terms in the numerators and denominators. Take care that you cancel the factors and not the terms.
- Finally, rewrite the remaining expressions.

Below are the few examples which will better explain the dividing rational expression technique.

*Example 1*

[(x^{2}Â + 3x – 28)/ (x^{2}Â + 4x + 4)]Â Ã· [(x^{2}Â – 49)/ (x^{2}Â – 5x- 14)]

__Solution__

=Â (x^{2}Â + 3x – 28)/ (x^{2}Â + 4x + 4)]Â Ã· [(x^{2}Â – 49)/ (x^{2}Â â€“ 5x – 14)

Factor both the numerators and denominators of each fraction.

âŸ¹ x^{2}Â + 3x – 28Â =Â (x – 4) (x + 7)

âŸ¹ x^{2}Â + 4x + 4Â =Â (x + 2) (x + 2)

âŸ¹ x^{2}Â – 49Â =Â x^{2}Â – 7^{2}Â =Â (x – 7) (x + 7)

âŸ¹ x^{2}Â â€“ 5x – 14Â =Â (x – 7) (x + 2)

=Â [(x – 4) (x + 7)/ (x + 2) (x + 2)]Â Ã· [(x -7) (x + 7)/ (x – 7) (x + 2)]

Now, multiply the first fraction by the reciprocal of the second fraction.

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

On cancelling common terms and rewrite the remaining factors to get;

=Â (x – 4)/ (x + 2)

*Example 2*

Divide [(2t^{2 }+ 5t + 3)/ (2t^{2 }+7t +6)]Â Ã·Â [(t^{2 }+ 6t + 5)/ (-5t^{2 }â€“ 35t – 50)]

__Solution__

Factor the numerators and denominators of each fraction.

âŸ¹ 2t^{2Â }+ 5t + 3Â =Â (t + 1) (2t + 3)

âŸ¹ 2t^{2Â }+ 7t + 6Â =Â (2t + 3) (t + 2)

âŸ¹ t^{2Â }+ 6t + 5Â =Â (t + 1) (t + 5)

âŸ¹ -5t^{2 }â€“ 35t -50Â =Â -5(t^{2}Â + 7t + 10)

=Â -5(t + 2) (t + 5)

=Â [(t + 1) (2t + 3)/ (2t + 3) (t + 2)]Â Ã·Â [(t + 1) (t + 5)/-5(t + 2) (t + 5)]

Multiply by the reciprocal of the second rational expression.

=Â [(t + 1) (2t + 3)/ (2t + 3) (t + 2)]Â *Â [-5(t + 2) (t + 5)/ (t + 1) (t + 5)]

Cancel common terms.

=Â -5

*Example 3*

[(x + 2)/4y]Â Ã·Â [(x^{2}Â – x – 6)/12y^{2}]

__Solution__

Factor the numerators of the second fraction

âŸ¹ (x^{2}Â – x – 6) =Â (x – 3) (x + 2)

=Â [(x + 2)/4y]Â Ã·Â [(x – 3) (x + 2)/12y^{2}]

Multiply by the reciprocal

=Â [(x + 2)/4y]Â *Â [12y^{2}/ (x – 3) (x + 2)]

On cancelling common terms, we get the answer as;

=Â 3y/4(x – 3)

*Example 4*

Simplify [(12y^{2}Â â€“ 22y + 8)/3y] Ã· [(3y^{2}Â + 2y – 8)/ (2y^{2 }+ 4y)]

__Solution__

Factor the expressions.

âŸ¹ 12y^{2}Â â€“ 22y + 8Â =Â 2(6y^{2}Â â€“ 11y + 4)

=Â 2(3y – 4) (2y – 1)

âŸ¹ (3y^{2}Â + 2y – 8) =Â (y + 2) (3y – 4)

= 2y^{2 }+ 4y =Â 2y (y + 2)

=Â [(12y^{2}Â â€“ 22y + 8)/3y] Ã· [(3y^{2}Â + 2y – 8)/ (2y^{2 }+ 4y)]

=Â [2(3y – 4) (y – 1)/3y] Ã· [y + 2) (3y – 4)/2y (y + 2)]

=Â [2(3y – 4) (2y – 1)/3y]Â *Â [y (y + 2)/ (y + 2) (3y – 4)]

=Â 4(2y – 1)/3

*Example 5*

Simplify (14x^{4}/y) Ã·Â (7x/3y^{4}).

__Solution__

=Â (14x^{4}/y)Â Ã·Â (7x/3y^{4})

=Â (14x^{4}/ y) * (3y^{4}/7x)

=Â (14x^{4Â }*Â 3y^{4}) / 7xy

=Â 6x^{3}y^{3}