Solving Rational Functions
Objective
Finding the Least Common Multiple
So here’s how we add fractions.
Let’s start off with an example:
We can’t just add these right?
We have no idea how to add them.
However, what if we changed them to something we can add?
But, we can’t change the equation right?
So, how do we change the numbers, without changing the amount?
Well, we can try to multiply the fractions by 1, but what kind of one?
This is where the Least Common Multiple comes in.
So, to start, let’s see which multiple the two denominators share.
To find it, let’s list out the multiples of 2:
2 4 6 8 10 12 14 16 18 20
Now, let’s list out the multiples of 3:
3 6 9 12 15 18 21 24 27 30
Now, which number do they both share?
6
6
Now we have to find out how to make 3 into 6.
We multiply it by 2!
Now we have to find out how to make 2 into 6.
We multiply it by 3!
THE CHEATING WAY
So, there is actually another way of creating a common denominator instead of finding the least common multiple
But, it’s not the best way.
However, it will work every single time.
We multiply by opposite denominators.
Sounds weird, but let’s try it:
So, what we do, is look at:
( )
= 35
Now, we multiply each side by the opposite number to get 35:
Again, this works every time, but it can take more work.
SO…….WHAT DOES THIS HAVE TO DO WITH ALGEBRA 2?
Again, this is not to insult your intelligence.
The truth is, you probably haven’t needed to find the least common denominator for years.
So, a little refresher on how to do it isn’t that bad of a thing.
But why do we need it?
Well, mainly because we are going to be adding fractions…….
With variables.
So, for example:
EXAMPLE 1
Add the following:
Alright, so this one seems a little weirder than what we’ve done before.
However, we can still do this.
First things first, we can’t add them together the way they are.
Mainly because we need a least common denominator.
Well, what we have so far is just x and 2.
Is there a possible way for us to maybe use the cheat way to find the LCD?
Well sure!
So, to use the cheat way, let’s first multiply the left side by 2/2:
And the right side by x/x:
Now, if we multiply the left side, we get:
And if we multiply the right side, we get:
Now we can add them together!
So, adding them together we get:
THAT WASN’T SO BAD, WAS IT?
Pretty easy right?
Well, it does get a little more complicated.
However, we’re usually using the cheat method to find the LCD.
So, let’s try a few more to make sure you got it.
EXAMPLE 2:
Add the following:
Again, we can’t seem to add these together the way they are.
Mainly because we need a least common denominator.
Well, what we have so far is x + 1 and x.
Is there a possible way for us to maybe use the cheat way to find the LCD?
Well sure!
However, it gets a little complicated.
First, we need to multiply the left side by x/x:
And the right side by x+1/x+1:
Now, if we multiply the left side, we get:
And if we multiply the right side, we get:
Now we can add them together!
So, adding them together we get:
And that’s our answer!
Now let’s try a little harder one:
Example 3:
Add the following:
Again, we can’t seem to add these together the way they are.
Mainly because we need a least common denominator.
Well, what we have so far is x + 1 and x - 2.
Is there a possible way for us to maybe use the cheat way to find the LCD?
Well sure!
However, it gets a little complicated.
First, we need to multiply the left side by x - 2/x - 2:
And the right side by x + 1/x + 1:
Now, if we multiply the left side, we get:
And if we multiply the right side, we get:
Now we can add them together!
So, adding them together we get:
And that’s our answer!
So, now, what if there is something harder?
But we don’t really want to multiply by a binomial?
Something like:
Example 4:
Add the following:
Again, we can’t seem to add these together the way they are.
Mainly because we need a least common denominator.
However, do we really want to multiply this monster by anything to make it even bigger?
Or, maybe there’s a way to make it smaller?
Well, can we factor this?
In fact we can!
When we factor the left side, we see that:
And when we factor the right side, we see:
Now that we see we have some common factors, we can see how easy we can get a LCD.
All we need to do is multiply the right side by -4/-4, so we can get:�12(x + 13) on both sides.
So:
And now we add, and we get:
Example 5:
Add the following:
Again, we can’t seem to add these together the way they are.
Mainly because we need a least common denominator.
However, do we really want to multiply this monster by anything to make it even bigger?
Or, maybe there’s a way to make it smaller?
Well, can we factor this?
In fact we can!
When we factor the left side, we see that:
And when we factor the right side, we see:
Now that we see we have some common factors, we can simplify some.
And what we’re left with is:
Adding what’s left, we get:
So now, how do we multiply rational functions?
Well, honestly there are two ways:
We can just multiply them together and hope for the best, then simplify it later.
Orrrrr……
We can simplify what we have and the multiply.
To do this, however, we need to review over
FACTORING
EXAMPLE 1:
Multiply the following:
Now, again, we could just multiply and then simplify
But dividing a 4th exponential polynomial by another 4th exponential polynomial may be a little bit of a nightmare.
So instead, let’s see if we can factor this and make it more simple.
So, now substituting in our factored expressions, we have:
Now we can cancel!
Lastly, we need to determine the excluded values of the expression.
Going back to the original expression, we saw that we had:
Setting up the denominators to 0, we see that:
x + 2 = 0 x – 4 = 0 x – 5 = 0
-2 -2 +4 +4 +5 +5
x = -2 x = 4 x = 5
Example 2:
Multiply the following:
Again, we need to factor the expression that we have to make it easier to multiply it.
So:
So, now substituting in our factored expressions, we have:
Now we can cancel!
And what we have left over is:
Simplifying our fraction, we finally get:
Lastly, we need to determine the excluded values of the expression.
Going back to the original expression, we saw that we had:
x + 3 = 0 x + 5 = 0 x + 8 = 0
+3 +3 -5 -5 -8 -8
x = 3 x = -5 x = -8
So, how do we divide rational functions?
To be blunt,
LITERALLY THE SAME WAY WE MULTIPLIED THEM
You factor first
Then you divide
However, when you divide fractions, remember, you have to first flip it, then multiply.
But again, it’s easier to show than to explain:
Example 1:
Divide the following:
First, just like when we were multiplying, we need to factor everything out so we can make sure to simplify.
So:
So, plugging in what we’ve factored, we have that:
Now that we’ve plugged in the factored form, we need to just flip it
Then multiply
Lastly, we need to determine the excluded values of the expression.
Going back to the original expression, we saw that we had:
Setting up the denominators to 0, we see that:
-7 -7 -2 -2
x = -7 x = -2 x = 0
Example 2:
Divide the following:
First, just like when we were multiplying, we need to factor everything out so we can make sure to simplify.
So:
So, plugging in what we’ve factored, we have that:
Now that we’ve plugged in the factored form, we need to just flip it
Then multiply
Lastly, we need to determine the excluded values of the expression.
Going back to the original expression, we saw that we had:
Setting up the denominators to 0, we see that:
+30 +30 - 1 - 1 +4 +4
3x = 30 x = - 1 x = 4
_______
3 3
x = 10
OKAY, SO HOW DO WE SOLVE RATIONAL EQUATIONS?
Well, basically we use what we’ve learned when it comes to expressions, and apply it to equations.
But, just like with solving rational expressions, it’s easier to show than to explain.
EXAMPLE 1:
Solve each rational equation algebraically:
So, first things first, we need to find the LCD.
Since we know the LCD of 3, 8, and 24 is 24.
And two of them need an x, so our LCD is 24x:
And we get:
Now, we subtract, and we get:
Now, we multiply by our denominators, and we get:
And we get:
EXAMPLE 2:
Solve each rational equation algebraically:
So, first things first, we need to find the LCD.
Since we know the LCD of 13, and x+3 is (13(x+3)).
And two of them need an x, so our LCD is (13(x+3)):
And we get:
Now, we add, and we get:
Now, we multiply by our denominators, and we get:
And we get:
-42x -42x
_____________
-37 -37