Rational expressions

Rational expressions are those involving ration of two polynomials or division of two polynomials. Rational expression: P(x)/P(y). Here P(x) and P(y) are polynomials. Here Q(x) cannot be zero.

Dividing Rational Expressions include division of two rational-expressions each of the form given above such that denominators of both the expressions are not zero.

Let us see How to Divide Rational Expressions now:

It would be easy for you to first understand division of rational numbers first before going to division of rational-expressions. To divide a rational number p/q by r/s we do multiplication of p/q and s/r. Actually, we need to reciprocate one of the numbers (second number) and then multiply it with the other number. Multiplication of rational numbers is done by simply multiplying the numerator by numerator and denominator by denominator.

Example: divide 4/3 by 9/5.
Solution: reciprocate the number 9/5 to get 5/9 and multiply it with 4/3.
4/(3 )÷9/5
 =  4/3.5/9
 =  ((4).(5))/((3).(9) )
 =  20/27

We divide rational form of expressions in the similar way as we divide rational numbers. Just reciprocate the second number and multiply it with the first number. Also check if the numerators  and denominators have any factors. If they have then write them in that form and try to cancel out the common factors of numerator and denominator.

If first rational-expression is f(x) = P(x)/Q(x) and second is g(x) = M(x)/ N(x), then f(x)/ g(x):
 = (P(x))/(Q(x))  (N(x))/(M(x)). here we have reciprocated g(x) and multiplied it with f(x).
 = (P(x)  .  N(x))/(Q(x)  .  M(x))

Let us see some examples to Divide Rational Expressions so that we are able to perform divisions of rational-expressions easily without any problem:

Example 1) divide  (y + 5)/3y by  (y+5)/(9y^2 ).
Solution 1) As shown in above explanation, just reverse the rational expression (y+5)/(9y^2 ) and multiply it with (y + 5)/3y :
=  (y + 5)/3y. (9y^2)/(y + 5)
= now here we see that y + 5 term is common in denominator and numerator so cancel it out. Also 3y is common on 3y and 9y2, so cancel one y also to get:
= 1/1. (3 y)/(y + 5)
= (3 y)/(y + 5)

Example 2) Divide (x^2+ 2x-15)/(x^2-4x-45)  by  (x^2+ x-12)/(x^2- 5x-36).
Solution 2) we can write (x^2+ 2x-15)/(x^2-4x-45)  ÷  (x^2+ x-12)/(x^2- 5x-36) as:
 (x^2+ 2x-15)/(x^2-4x-45).  (x^2- 5x-36)/(x^2+ x-12)
= ((x +5)(x - 3))/((x - 9)  (x + 5)).((x - 9)(x + 4))/((x + 4)  (x - 3))
See that all the terms are cancelling out in numerator with denominator. So we get
 (x^2+ 2x-15)/(x^2-4x-45)  ÷  (x^2+ x-12)/(x^2- 5x-36) = 1