Welcome to Georgia Highlands College Math 97 and Math 99 video tutorials.
In this video segment we’ll be answering the question,
how do you subtract rational expressions with unlike denominators.
So the first step you’re going to do is just like addition except when you get
to the part where you usually add them, you're going to be subtracting them and
distributing that negative one.
So you find the LCD first. You write equivalent expressions with the LCD as the
denominator, and you do that by multiplying by the clever form of one, subtract
your numerators over your LCD. And this means you’re going to have to distribute
that negative one once you’ve subtracted. Combine any like terms in the
numerator and finish it off by factoring and simplifying.
Let’s take a look at an example.
All right, so I have the example here. Five over Y squared minus 5Y minus Y over
5Y minus 25.
So the first step in this process is to find the LCD. So I just need to factor
each of these denominators, Y squared minus 5Y has a common factor of Y and when
I divide that out I’m left with Y minus 5 and 5Y minus 25 has a GCF of 5 and
when I factor that I'm left with Y minus 5.
So the next step in finding the LCD is to list out the factors of the first
denominator. So Y times Y minus 5. And add any factors in the second denominator
that aren’t already in the list. So we’ve already got a Y minus 5, so we don't
need that, but we do need an extra factor of 5.
So our LCD for this problem is, we’ll actually put the 5 out front, just to keep
it in good form. 5Y times Y minus 5. So we have three factors here 5, Y, and Y
minus 5.
All right, so moving back to the problem, we want to go ahead and write the
original problem with the denominators factored. So Y times Y minus 5 and 5
times Y minus 5.
And give yourself plenty of room to come back and make equivalent fractions and
to do that we multiply by one in the form of whatever factor over whatever
factor we need to make the LCD.
So with this first expression we need a five or a factor of five in there and
with the second expression, we need a factor of Y.
All right, moving along, you actually do the multiplication. So there and there,
and put it over what is now your common denominator.
So this first expression gives 25/5Y times Y minus 5 minus Y times Y is Y
squared over 5Y times Y minus 5. And putting our numerators over our common
denominator we’re subtracting the numerators. So 25 minus, now normally, you’d
want to use parentheses here because you may have a polynomial that that
negative needs to distribute to, so I'll just go ahead and write the Y squared
in parentheses there. And that's over our common denominator of 5Y times Y minus
5.
So there is an understood negative 1 that gets multiplied with that Y squared
giving us 25 minus Y squared over 5 times Y times Y minus 5.
5.
All right, bells and whistles should be going off because this numerator is not
in standard form. When we rearrange it into standard form you may notice, hey,
that's beginning to look like a difference of squares. It's not quite a
difference of squares yet, but if we simply factor out a negative 1, we are left
with a difference of squares. Y squared minus 25/5 times Y times Y minus 5.
All right, so since that binomial in the numerator is the difference of squares,
it can be factored as Y plus times Y minus 5. That's all over our common
denominator of 5 times Y times Y minus 5. So we do have common factors that
divide out because anything divided by itself is one.
You can see that we have a positive, I’m sorry, a negative being divided by a
positive, so that’s just going to make our entire expression negative. I can
just bring that negative one out front to multiply with the entire expression
and we have left in the numerator Y plus 5 and in the denominator, 5Y.
Now it may be very tempting to come back and divide these Y’s out. You cannot do
that. That is not correct because the Y in the numerator is a term and when you
are dividing those factors out, you only divide factors, never divide terms. So
that’s your final answer, negative Y plus 5 over 5Y.
So I hope that this has been helpful for you it. It was definitely a complex
example in looking at how to subtract rational expressions with unlike
denominators.
So a real quick recap. Find the LCD, write equivalent expressions multiplying by
one, subtract the numerators over the LCD, don't forget to distribute that
negative, combine your like terms, factor, and simplify.
If you have any other questions about adding, subtracting, multiplying,
dividing, anything about rational expressions, make sure to contact your
Highlands instructor.
Thank you.
