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# Dividing Expressions – Methods & Examples

An algebraic expression is a mathematical phrase where variables and constants are combined using the operational (+, -, × & ÷) symbols. For example, 10x + 63 and 5x – 3 are examples of algebraic expressions.

A rational expression is simply defined as a fraction in either or both the numerator and the denominator is an algebraic expression. Examples of rational fractions are: 3/ (x – 3), 2/ (x + 5), (4x – 1)/3, (x^{2} + 7x)/6, (2x + 5)/ (x^{2} + 3x – 10), (x + 3)/(x + 6) etc.

## How to divide the ordinary fractions?

Rational expressions are divided by applying the same steps used to divide ordinary fractions having rational numbers. A rational number is a number that is expressed in the form of p/q; where ‘p’ and ‘q’ are integers and q ≠ 0. In other words, a rational number is simply a fraction where the integer a is the numerator and integer b is the denominator.

*Example of rational numbers include:*

2/3, 5/8, -3/14, -11/-5, 7/-9, 7/-15 and -6/-11 etc.

Division of ordinary fractions is done by multiplying the first fraction by the reciprocal of the second fraction. For example, to divide, 4/3 ÷ 2/3, you simply find the product of the first fraction and the inverse of the second fraction; 4/3 x 3/2 = 2.

*Other examples of dividing rational numbers are:*

9/16 ÷ 5/8 = 9/16 × 8/5

= (9 × 8)/ (16 × 5) = 72/80

= 9/10

-6/25 ÷ 3/5 = -6/25 × 5/3

= {(-6) × 5}/(25 × 3)

= -30/75

= -2/5

## How to Divide Rational Expressions?

Similarly, when dividing rational expressions, we invert or flip the second expression and multiply it by the first expression.

*Below is a summary of the steps followed when dividing rational expressions:*

- Completely factor out the denominators and numerators of all expressions.
- Replace the division sign (÷) with the multiplication sign (x) and find the reciprocal of the second fraction.
- Reduce the fraction if possible.
- Now rewrite the remaining factor.

* *

*Example 1*

Divide 4x/3 ÷ 7y/2

__Solution__

4x/3 ÷ 7y/2 = 4x/3 * 2/7y

=8x/21y

* *

*Example 2*

Divide ((*x* + 3) / 2x^{2}) ÷ (4 / 3x)

__Solution__

Change the division sign to multiplication sign and invert the second expression;

= (*x* + 3 / 2x^{2}) × (3x/ 4)

Multiply the numerators and denominators separately if they cannot be factored out;

= [(*x* + 3) × 3x] / (2x^{2} × 4)

= (3x^{2} + 9x) / 8x^{2}

Since there is a common factor of x in both the numerator and the denominator, therefore, this expression can be simplified as;

(3x^{2} + 9x) / 8x^{2} = *x* (3x+ 9) / 8x^{2}

= (3x + 9) / 8x

* *

*Example 3*

Divide and then simplify.

(x ^{2} − 4)/ (x + 6) ÷ (x + 2)/ (2x + 12)

__Solution__

Multiply the first expression by the reciprocal of the second expression;

The reciprocal of the second fraction (x + 2)/ (2x + 12x) is (2x + 12x)/ (x + 2)

(x ^{2} − 4)/ (x + 6) ÷ (x + 2)/ (2x + 12) = (x ^{2} − 4)/ (x + 6) * (2x + 12x)/(x + 2)

= Now multiply the numerators and the denominators.

= [(x^{2} − 4) (2x + 12)]/ [(x + 6) (x + 2)]

Factor the terms in the numerator and cancel out the common factors

= [(x + 2) (x − 2) * 2(x + 6)]/ (x + 6) (x + 2)

Rewrite the remaining fraction;

=2(x − 2)/1= 2x−4

* *

*Example 4*

Divide (*x* + 5) / (*x* – 4) ÷ (*x* + 1)/x

__Solution__

Find the reciprocal of the second expression;

Reciprocal of (*x* + 1)/x = x/x + 1

Now multiply the fractions;

= ((*x* + 5) * *x*) / ((*x* – 4) * (*x* + 1))

= (*x*^{2} + 5x) / (*x*^{2} – 4x + *x* – 4)

= (*x*^{2} + 5x) / (*x*^{2} – 3x – 4)

* *

*Example 5*

Simplify {(12x – 4x ^{2})/ (x ^{2} + x – 12)} ÷ {(x ^{2 }– 4x)/ (x ^{2} + 2x – 8)}

__Solution__

Invert the second fraction and multiply;

= {(12x – 4x ^{2})/ (x ^{2} + x – 12)} *{(x ^{2} + 2x – 8)/ (x ^{2 }– 4x)}

Factor out both the both the numerators and denominators of each expression;

= {- 4x (x – 3)/(x-3) (x + 4)} * {(x – 2) (x + 4)/x (x + 2) (x – 2)}

Reduce or cancel the expressions and rewrite the remaining factors;

= -4/ x + 2

** **

*Practice Questions*

Simplify the following rational expressions:

- 2x/4y ÷ 3y/4xy
^{2} - (8x
^{2 }– 6x/ 4 – x) ÷ (4x^{2}-x – 3/ x^{2}-16) ÷ (2x + 8/-5x -5). - (x
^{2 }– 7x + 10/ x^{2 }– 9x + 14) ÷ (x^{2}+ 6x + 5/ x^{2 }– 6x -7) - (2x + 1/x
^{2}– 1) ÷ (2x^{2}+ x/ x + 1) - (-3x
^{2}+27/x^{3 }– 1) ÷ (x – 3x/7x^{3}+ 7x^{2}+ 7x) ÷ (21/x – 1) - (x
^{2}– 5x – 14/ x^{2}– 3x + 2) ÷ (x^{2}– 14x + 49/ x^{2}– 4) - When (4x + 55) is divided by (2x + 3), the result is 9. Find the value of x.

** **

__Answers__

- 2x
^{2}/3 - 5x
- x+2/x-2
- 1/x (x – 1)
- – x – 3
- (x + 2)
^{2}/ (x – 1) (x – 7) - 2

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