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Simplify Rational Expressions by Factoring

Rational Expressions are just like the big brother to fractions. They have this semi elaborate name but really all they are is a special fraction where the numerator or denominator contains expressions rather than just numbers. So while learning how to simplify rational expressions can at times seem scary, in their basic form they are just fractions. We are going to apply properties we are already aware of so it will be no sweat once we start learning and include a little practice.

So what are these properties we know or are supposed to know?

1.) The division property states that when a number, variable or expression is divided by itself, it is equal to one

$frac{3}{3} = 1$

I will use a basic example first because most people can relate to this problem by rewriting the problem as a multiplication problem 3*1=3 making our point that three divided by three is equal to one. When we have variables where we do not know the value such as x, we can also apply the division property as:

$frac{x}{x} = 1$

because again x can represent any number so any number divided by itself is one. Lastly the division property also works when with expressions such as:

$frac{3x}{3x} = 1$ and $frac{x+3}{x+3} = 1$

To understand why these work we replace x with any number. I will use 5.

$frac{3(5)}{3(5)} = frac{15}{15} =1$ and $frac{(5)+3}{(5)+3} = frac{8}{8} =1$

The division property also works across multiplication but not addition and subtraction. What I mean by that is that we can apply the division property when we have two expressions or terms multiplied by each other but not when they are added or subtracted.

$frac{7x}{x} = 7$ and $frac{7(x+1)}{7} = x+1$

because
$frac{7(5)}{(5)} = frac{35}{5} =7$ and $frac{7(5+1)}{7} = frac{42}{7} = 6$
However
:
$frac{x+3}{3} =x$ because $frac{5+3}{3} = frac{8}{3} $ not 5

2.) Factoring expressions is the process of rewriting an expression as the product of its factors. Factors are numbers, terms or expressions that evenly divide into an expression by another factor. There are multiple different ways to factor an expression but for this series we will focus on factoring out a GCF as well as factoring trinomials. Examples of factoring an expression that by GCF are:

$3x+6=3(x+2)$ and 13x^2-3x=3x(x-1)

An example of factoring out a trinomial is

$x^2-3x-4=(x+1)(x-4)$

If you would like detailed explanation on how to factor please see my blog post on factoring expressions.

Simplifying a rational expression can really be broken down to just applying the division property to a fraction where there are expressions in the numerator and denominator. However the problems you will more than likely encounter are not going to be in a form where we can simply apply the division property. We will have to factor the expressions to separate the expression by multiplication then we can then apply the division property to simplify.

$frac{x^2+x-6}{x+3} $ can be factored to $frac{(x+3)(x-2)}{(x+3)} $

Now that (x+3)and (x-2) are separated by multiplication so we can apply the division property to simplify and divide out the (x+3) to one.

So what does this allow us to do? Well from the beginning of your journey of learning math we are always looking to make more complicated problems simple. As we move on to adding, subtracting, multiplying, dividing, rational expressions as well as solving and graphing rational equations, we will have some very complicated rational expressions and equations to work with. If we can simplify first we will reduce the number of steps as well as work with a problem that is in a simpler form.

For more examples on how to simplify rational expressions please check out my free tutorials on my website and if you are of need of a more comprehensive learning experience. I have included the link to my Udemy course with a special 50% off coupon. If you have any questions please submit below.