What is the easiest way to simplify rational expressions?
Currently, this is the area I am having the most trouble in; I want to be able to understand this better before my final next week. Any help would be appreciated.
You need to focus on two chapters to understand how to simplify rational expressions.
The first chapter contains the addition/subtraction of rational expressions that do not have a common denominator.
Since you will understand better an worked example, you may consider the following example such that:
`1/(x-1) + (2x-5)/(x+1)`
Notice that the fractions do not have a common denominator, hence, you will need to identify the common denominator. In this case, since the denominators do not share a common factor, the common denominator will be the following product, such that:
`(x+1)(x-1) = x^2- 1`
You need to identify what is the missing factor to the first denominator and to multiply the fraction by the value identified, such that:
`1*(x+1)/((x-1)(x+1)) + ((2x-5)(x-1))/((x-1)(x+1)) `
Since the fractions share the same denominator, you may write the expression such taht:
`(x + 1 + (2x - 5)(x - 1))/(x^2 - 1)`
You need to open the brackets such that:
`(x + 1 + 2x^2 - 2x - 5x + 5)/(x^2 - 1) = (2x^2 - 6x + 6)/(x^2-1)`
Factoring out 2 to numerator yields:
`(2x^2 - 6x + 6)/(x^2-1) = 2(x^2 - 3x + 3)/(x^2-1)`
Hence, simplifying the expression yields `1/(x-1) + (2x-5)/(x+1) = 2(x^2 - 3x + 3)/(x^2-1).`
The first chapter contains the multiplication or division of rational expressions.
Selecting the following example yields:
You need to identify if there exists duplicate factors to numerator and denominators and cancel the factors, such that:
`1/(x+1)*(x-1)/1` ( notice that `5x-1` is the duplicate factor)
Hence, simplifying the expression yields `(5x-1)/(x+1)*(x-1)/(5x-1) = (x-1)/(x+1).`