Cancelling common factors
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Given a rectangle with length a-squared plus 6a minus 27 and a width a-squared minus 9, Write the ratio of the width of the rectangle to its length as a simplified rational expression. So we want the ratio of the width to the length to the length of the rectangle, and they give us the expressions for each of these. The width—the expression for the width of the rectangle—is a-squared minus 9. So, the width —let me do it in this pink—is a-squared minus 9, and we want the ratio of that to the length; the ratio of the width of the rectangle to its length. The length is given right over there; it is a-squared plus 6a minus 27, they want us to simplify this. And so the best way to simplify this, whether we're dealing with expressions in the numerator or denominator, or just numbers, is we want to factor them and see if they have common factors and if they do we might be able to cancel them out. So if we factor this top expression over here—which was the expression for the width—this is of the form a-squared minus b-squared, where b-squared is 9. So this is going to be the same thing as a plus the square root of 9 times a minus the square root of 9. So this is a plus 3 times a minus 3 and I just recognized that from just the pattern; if you ever see something a-squared minus b-squared, it's a plus b times a minus b and you can verify that for yourself; multiply this out, you'll get a-squared minus b-squared. So, the width can be factored into a plus 3 times a minus 3. Let's see if we can do something for the denominator. So here, if we want to factor this out, we have to think of two numbers that when we add them, I get positive 6, and when I take their product, I get negative 27. Let's see, if I have positive 9 and negative 3, that would work. So, this could be factored as a plus 9 and a minus 3. 9 times a is 9a, a times negative 3 is negative 3a, when you add those two middle terms together, you'll get 6a, just like that and then, 9 times negative 3 is negative 27—of course the a times a is a squared. So I've factored the two expressions and let's see if we can simplify it. And before we simplify it, because when we simplify it we lose information, let's just remember what are allowable a's here, so we don't lose that information. Are there any a values here that will make this expression undefined? Well, any a value that makes the denominator zero will make this undefined. So a cannot be equal to negative 9 or 3, 'cause if a was negative 9 or 3, then the denominator would be zero; this expression would be undefined. So we have to remember this, this is part of the expression; we don't want to change this domain; we don't want to allow things that weren't allowable to begin with, so let's just remember this right over here. Now with that said, now that we've made this constraint, we can simplify it more; we can say, look we have an a minus 3 in the numerator and we have an a minus 3 in the denominator and we're assuming that a is not going to be equal to 3, so it's not like we're dividing zero over zero. So a will not be equal to 3, any other number; this will be an actual number, you divide the numerator and denominator by that same value, and we are left with a plus 3 over a plus 9 and the constraint here—we don't want to forget the constraints on our domain— a cannot equal negative 9 or 3. And it's important that we write this here, because over here we lost the information that a could not be equal to 3, but in order for this to really be the same thing as this thing over here, when a was equal to 3 it wasn't defined so in order for this to be the same thing, we have to constrain the domain right over there; a cannot be equal to 3. Hopefully you found that useful.