Welcome to Georgia Highlands College Math 97 and Math 99 video tutorials.
In this video segment we’ll be answering the same question that we answered in
the last video. However, we’ll look at a more complex example of it.
So how do you simplify complex rational expressions? Notice I have a 2 there
because it’s the second video in the series.
Well if needed, you’re to start out by adding or subtracting to get a single
rational expression in the numerator and then in step two you’re going to repeat
that same process for the denominator. But what it boils down to is that you're
going to work with the complex rational expression that you’ve been given until
you can get it into single rational expression divided by single rational
expression here.
And then, once you have it in this form, well this is just a division, indicated
by the big fraction bar here, a division of rational expressions. And we know
from earlier videos that we never divide rational expressions, we simply
multiply by the reciprocal of the divisor.
So that's what we’ll employ after we have it in this form, and then finally we
factor and simplify if possible.
Let’s go ahead and take a look at an example.
All right, so we have an example here of a complex rational expression and we
will begin by taking that first step of getting the numerator into a single
rational expression.
So we are subtracting these rational expressions in the numerator so the first
thing we need to do is factor that second denominator. So, through the magic of
technology, I have already factored that denominator and it factors as X plus 5
times X minus 2.
Now if you’re not sure how I got that, you can always go back and watch any of
the videos on factoring.
So that's my first step there. And I’m going to go ahead and write this 1 that’s
leading in my denominator as 1/1. I just want to make everything look like
rational expressions so I don't get confused.
And I'll go ahead and put parentheses around that X minus 2 in this one here,
just to remind myself that these are factors themselves with two terms.
So, moving along, we’re still trying to get that numerator and denominator as a
single rational expression, so to do so in the numerator we have to find that
LCD.
Well the first rational expression is missing an entire factor of X plus 5, so
I'm just going to multiply by 1 in the form of X plus 5 over X plus 5, which was
also discussed in a previous video.
And if you look at the denominator, that first rational expression is missing a
factor of X minus 2. So I will multiply by 1, which does not change anything, it
just makes it look different here with the clever form of 1 that I’ve chosen to
multiply by X minus 2 over X minus 2.
Now I'm ready to distribute and see what I get. So X times 1 is X, 5 times 1 is
positive 5 and that is over my now common denominator for my numerator, X plus 5
times X minus 2. And I'm going to subtract 6 over X plus 5 times X minus 2, and
all of that will be divided by, well I have X minus 2 over X minus 2 now in my
numerator of my denominator, my first numerator, and I'm going to be adding that
to 1 over X minus 2.
Okay, so now we've gotten to the point were ready to start putting these
rational expressions together in the numerator and denominator. So to subtract
rational expressions, we subtract the numerators X plus 5 minus 6 over the
common denominator of X plus 5 times X minus 2 and moving to the big
denominator, I’m just going to make that fraction bar a little more prominent,
we have X minus 2 plus 1 divided by X minus 2.
All right, so I can combine like terms in both of my numerators and I end up
with X minus 1 over X plus 5 times X minus 2 big divided by, and once again a
big prominent division bar there, X minus 1 over X minus 2.
All right, so I’m going to come back and show that there's a factor there,
there's a factor there and there.
All right, so now I've finally gotten to the point where it looks like single
rational expression divided by a single rational expression. So I'm just
dividing rational expressions here at this point, which once again we never
divide rational expressions we multiply by the reciprocal of the divisor. So I
keep my numerator X minus 1 divided by X plus 5 times X minus 2. And now I'm
going to change that, so that I am multiplying by the reciprocal of the divisor
which is X minus 2 over X minus 1, all right.
So, our next step in the multiplication process for rational expressions is just
to express it as a single rational expression, multiplying numerator with
numerator and denominator with denominator.
I can divide any common terms out because anything divided by itself is one,
anything divided by itself is one, and don't forget, if you cross the line,
leave one behind. So that numerator doesn't disappear, there is an understood
one being multiplied and it must show up in your final answer.
So your final rational expression is one over X plus 5. So that's what you get
when you simplify that complex rational expression.
I hope that this has been helpful for you in diving a little deeper into a more
complex, complex rational expression, and as usual, if you have any questions
about this process, about factoring, adding and subtracting rational
expressions, anything that showed up in this problem, please make sure to
contact your Highlands instructor.
Thank you.
