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Simplifying Rational Functions with Factoring and GCFs - Concept
Simplifying rational expressions combines everything learned about factoring common factors and polynomials. When simplifying rational functions, factor the numerator and denominator into terms multiplying each other and look for equivalents of one (something divided by itself). Include parenthesis around any expression with a "+" or "-" and if all terms cancel in the numerator, there is still a one there.
A lot of times you're going to be given a rational expression and asked to simplify it. Simplify if means you want to eliminate any factors that are the same on top and bottom of the fraction. So here are some things to keep in mind, first thing make sure you factor everything completely and then cross out any factors that are exactly the same on top and bottom. They have to be exactly the same, watch out for what we call opposite factors, there'll be something like this, if you have x minus 2 and 2 take away x, those are really close they're what we call opposite because this guy is actually that guy multiplied by negative 1. Let me prove it to you, if I were to factor out a negative 1 from this second factor parentheses grouping I would see negative 1 times negative 2 plus x which is the same thing a negative 1 times x minus 2.
Now you can see that those are what we call opposite factors, this guy just got multiplied by negative one. That's something that'll help you, when you're simplifying rational expressions. So again the most important thing is to factor everything completely and only cross out factors that are the same in top and bottom. That means it has to be in it's multiplied form, you can't cross out something like this, let me show you. If I had x plus 2 divided by 2 a lot of students would be tempted to cancel out those 2s but in fact that's not a factor. A factor means things have to be multiplied so on top that's not a factor like 2 is not a factor of this x plus 2 business. I could cancel out if I had instead of that, if I had 2x over 2 now those 2's are eligible to be reduced because 2 over 2 is the same thing as 1. So keep that in mind when you're simplifying rational expressions. Factor everything and then only cross out factors that are exactly the same on the top and bottom of your fraction.