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## College Algebra

### Course: College Algebra>Unit 8

Lesson 3: Reducing rational expressions to lowest terms

# Reducing rational expressions to lowest terms

Sal explains what it means to reduce a rational expression to lowest terms and why we would want to do that. Just don't forget the excluded values! Created by Sal Khan and CK-12 Foundation.

## Want to join the conversation?

• do you always have to add the condition? never heard of it from my teacher.
• Sometimes in an Algebra 1 course/text/curriculum, teachers will just teach the simplifying piece, and leave the restrictions for Algebra 2. This is because this is one of the most challenging type of problems in Algebra 1. Brains often melt solving rationals because many students can barely factor and simplify, let alone consider restrictions on the denominator. Even Sal makes mistakes in his examples, he forgets restrictions.
• Do we not also have to state that (x cannot equal 0.5) as well as (x cannot equal -3) for the final example, because this would also make the denominator equal to 0?
• No, you need not explicitly state such information. The reason being, that information is actually contained in the final answer of (3x-6)/(2x-1). The reason that x cannot equal -3 has already been removed from the problem by the time we write the final answer, so it must be explicitly stated. As a general rule, you need not state exceptions if they are obvious in the final answer. A time you may wish to state final answer exceptions is when you have distributed out the denominator.
• At around , shouldn't there also be the exception of x not equal to 2, because the value of 2 also makes the denominator 0.
• Well, not really. Notice that we list the condition that was cancelled out, because we can no longer see the factor of (x+1) in the denominator. Since we can still see the (x-2) factor, we can tell by looking at it that x can't =2, which is why it isn't listed. We only have to list those exclusions from the domain that are no longer visible.
• Near couldn't you just factor out a 3 and get 3(x^2+x-6) and then factor the rest of it using the quadratic formula or other methods?
• That would work if I'm not mistaken, so if you're comfortable with that, go ahead! I personally prefer the method Sal uses.
• why is it that all math problems have numbers in them?
• I guess an answer would be that numbers can represent an almost unlimited amount of things. I wouldn't be surprised if there were some math problems without numbers, but it is a very essential part of the subject and so it shows up in lots of places. Something to think about would be, if not numbers, what else would be in them?
• At , Sal said that x cannot equal -1. However, x also cannot be equal to 2, based off of the answer he got. Why wouldn't you put x cannot equal negative 2 in the final answer, as that would also make the expression undefined?
• Now that the expression has been simplified to (x+5)(x-2), it is obvious that x cannot be equal to 2, but the fact that x could also not be equal to -1 (otherwise division by 0 resulted) in the original non simplified expression has been lost, so we add the condition as a reminder.
You see, we can set x=-1 into (x+5)(x-2) without a problem, but we could not set x=-1 in the original expression - and this simplified expression is based on that.
• how do you get to the practice problems for this
• at , i still do not get how he got x cannot equal -1 over 3
• I've come across problems in my homework where I don't know what the condition should be. Is it the x value that makes what you cancel equal to zero, that makes the original expression's denominator equal to zero, or that makes the new expression's denominator equal to zero?
• In problems like those in the video you are expected to explicitly state the condition for the x value that makes what you cancel equal to zero.

However, since you're cancelling (and so there is such a factor in the original denominator), it implies the same thing makes the original expression's denominator equal to zero, or maybe—is ONE OF THE THINGS that make the original expression's denominator equal to zero.

If you think about it, the value that makes the new expression's denominator equal to zero is the reason for saying "one of the things..." in the previous sentence. Because THE OTHER THING is what makes the new expression's denominator equal to zero. It's the restriction (or several restrictions) shared by the old and the new expressions, therefore you don't have to explicitly state it in a task like this.