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# Dividing Rational Expressions – Techniques & Examples

Rational expressions in mathematics can be defined as fractions in which either or both the numerator and the denominator are polynomials. 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 division sign (÷) to multiplication sign (×).

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

- a/b ÷ c/d = a/b × d/c = ad/bc

For example;

- 5/7 ÷ 9/49 = 5/7 × 49/9

= (5 × 49)/ (7 × 9) = 245/63

= 35/9

- 9/16 ÷ 5/8

= 9/16 × 8/5

= (9 × 8)/ (16 × 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 cancelling 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} + 3x – 28)/ (x^{2} + 4x + 4)] ÷ [(x^{2} – 49)/ (x^{2} – 5x- 14)]

__Solution__

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

Factor both the numerators and denominators of each fraction.

⟹ x^{2} + 3x – 28 = (x – 4) (x + 7)

⟹ x^{2} + 4x + 4 = (x + 2) (x + 2)

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

⟹ x^{2} – 5x – 14 = (x – 7) (x + 2)

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

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

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

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

= (x – 4)/ (x + 2)

*Example 2*

Divide [(2t^{2 }+ 5t + 3)/ (2t^{2 }+7t +6)] ÷ [(t^{2 }+ 6t + 5)/ (-5t^{2 }– 35t – 50)]

__Solution__

Factor the numerators and denominators of each fraction.

⟹ 2t^{2 }+ 5t + 3 = (t + 1) (2t + 3)

⟹ 2t^{2 }+ 7t + 6 = (2t + 3) (t + 2)

⟹ t^{2 }+ 6t + 5 = (t + 1) (t + 5)

⟹ -5t^{2 }– 35t -50 = -5(t^{2} + 7t + 10)

= -5(t + 2) (t + 5)

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

Multiply by the reciprocal of the second rational expression.

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

Cancel common terms.

= -5

*Example 3*

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

__Solution__

Factor the numerators of the second fraction

⟹ (x^{2} – x – 6) = (x – 3) (x + 2)

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

Multiply by the reciprocal

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

On cancelling common terms, we get the answer as;

= 3y/4(x – 3)

*Example 4*

Simplify [(12y^{2} – 22y + 8)/3y] ÷ [(3y^{2} + 2y – 8)/ (2y^{2 }+ 4y)]

__Solution__

Factor the expressions.

⟹ 12y^{2} – 22y + 8 = 2(6y^{2} – 11y + 4)

= 2(3y – 4) (2y – 1)

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

= 2y^{2 }+ 4y = 2y (y + 2)

= [(12y^{2} – 22y + 8)/3y] ÷ [(3y^{2} + 2y – 8)/ (2y^{2 }+ 4y)]

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

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

= 4(2y – 1)/3

*Example 5*

Simplify (14x^{4}/y) ÷ (7x/3y^{4}).

__Solution__

= (14x^{4}/y) ÷ (7x/3y^{4})

= (14x^{4}/ y) * (3y^{4}/7x)

= (14x^{4 }* 3y^{4}) / 7xy

= 6x^{3}y^{3}

*Practice Questions*

Divide each of the following rational expressions:

- [(a + b)/ (a – b)] ÷ [(a³ + b³)/ [(a³ – b³)]
- [(x² – 16)/ (x² – 3x + 2)] ÷ [(x³ + 64)/ (x
^{2}– 4)] ÷ [(x² – 2x – 8)/ (x² – 4x + 16)] - [(x² – 4x – 12)/ (x² – 3x – 18)] ÷ [(x² + 3 x + 2)/ (x² – 2x – 3)]
- [(p² – 1)/p] [p²/ (p – 1)] ÷ [(p + 1)/1]
- [(2 x – 1)/ (x² + 2x + 4)] ÷[(2 x² + 5 x -3)/(x⁴ – 8 x)] ÷ [(x² – 2x)/ (x + 3)]

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