- Home
- >
- Dividing Expressions â€“ Methods & Examples

# Dividing Expressions â€“ Methods & Examples

An algebraic expression is a mathematical phrase where variables and constants are combined using the operational (+, -, Ã— & Ã·) 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}Â + 7x)/6, (2x + 5)/ (x^{2}Â + 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 â€˜pâ€™ and â€˜qâ€™ are integers and q â‰ 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 Ã· 2/3, you 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 Ã· 5/8 = 9/16 Ã— 8/5

= (9 Ã— 8)/ (16 Ã— 5)Â = 72/80

= 9/10

-6/25 Ã· 3/5 = -6/25 Ã— 5/3

= {(-6) Ã— 5}/(25 Ã— 3)

= -30/75

= -2/5

## How to Divide Rational Expressions?

Similarly, we invert or flip the second expression when dividing rational expressions 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 (Ã·) with the multiplication sign (x) and find the reciprocal of the second fraction.
- Reduce the fraction if possible.
- Now rewrite the remaining factor.

*Â *

*Example 1*

Divide 4x/3 Ã· 7y/2

__Solution__

4x/3 Ã· 7y/2 = 4x/3 * 2/7y

=8x/21y

*Â *

*Example 2*

Divide ((*x*Â + 3) / 2x^{2}) Ã· (4 / 3x)

__Solution__

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

= (*x*Â + 3 / 2x^{2}) Ã— (3x/ 4)

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

= [(*x*Â + 3) Ã— 3x] / (2x^{2}Â Ã— 4)

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

= (3x + 9) / 8x

*Â *

*Example 3*

Divide and then simplify.

(x ^{2}â€‰âˆ’â€‰4)/ (xâ€‰+â€‰6) Ã· (xâ€‰+â€‰2)/ (2xâ€‰+â€‰12)

__Solution__

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

The reciprocal of the second fractionÂ (xâ€‰+â€‰2)/ (2xâ€‰+â€‰12x) isÂ (2xâ€‰+â€‰12x)/ (xâ€‰+â€‰2)

(x ^{2}â€‰âˆ’â€‰4)/ (xâ€‰+â€‰6) Ã· (xâ€‰+â€‰2)/ (2xâ€‰+â€‰12) = (x ^{2}â€‰âˆ’â€‰4)/ (xâ€‰+â€‰6) * (2xâ€‰+â€‰12x)/(xâ€‰+â€‰2)

= Now multiply the numerators and the denominators.

= [(x^{2}â€‰âˆ’â€‰4) (2xâ€‰+â€‰12)]/ [(xâ€‰+â€‰6) (xâ€‰+â€‰2)]

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

= [(xâ€‰+â€‰2) (xâ€‰âˆ’â€‰2) * 2(xâ€‰+â€‰6)]/ (xâ€‰+â€‰6) (xâ€‰+â€‰2)

Rewrite the remaining fraction;

=2(xâ€‰âˆ’â€‰2)/1= 2xâˆ’4

*Â *

*Example 4*

Divide (*x*Â + 5) / (*x*Â â€“ 4) Ã· (*x*Â + 1)/x

__Solution__

Find the reciprocal of the second expression;

Reciprocal of (*x*Â + 1)/x = x/x + 1

Now multiply the fractions;

= ((*x*Â + 5) *Â *x*) / ((*x*Â â€“ 4) * (*x*Â + 1))

= (*x*^{2}Â + 5x) / (*x*^{2}Â â€“ 4x +Â *x*Â â€“ 4)

= (*x*^{2}Â + 5x) / (*x*^{2}Â â€“ 3x â€“ 4)

*Â *

*Example 5*

Simplify {(12x â€“ 4x ^{2})/ (x ^{2} + x – 12)} Ã· {(x ^{2 }– 4x)/ (x ^{2} + 2x – 8)}

__Solution__

Invert the second fraction and multiply;

= {(12x â€“ 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 â€“ 3)/(x-3) (x + 4)} * {(x â€“ 2) (x + 4)/x (x + 2) (x â€“ 2)}

Reduce or cancel the expressions and rewrite the remaining factors;

= -4/ x + 2