Dividing Rational Expressions
Objective
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
Factoring a quadratic
Example 1
Let’s say we are given:
We can see that it’s just 1, so there’s nothing to add to our factored equation.
However, we can see that we have 12 as our number.
So we break 12 into its multiples, and see if we can add two of those multiples together to get 7.
Well, we can see that
12 = 6 * 2
12 = 3 * 4
We know 6 + 2 = 8
But 3 + 4 = 7!
So, now we have our pieces!
Factoring the quadratic
So now we just plug in what we have:
So:
(x + )(x + )
And we know that 12 = 3 * 4, and 3 + 4 = 7, so:
(x + 3)(x + 4)
Now we solve!
(x + 3)(x + 4) = 0
x + 3 = 0
-3 -3
x = -3
And
x + 4 = 0
-4 -4
x = -4
So x = -3, -4!
So let’s try another one
Let’s say we’re given:
(x + )(x + )
And we know 2 = 2 * 1
And 2 + 1 = 3
So:
(x + 2)(x + 1) = 0
Now we solve!
x + 2 = 0
- 2 -2
x = -2
And
x + 1 = 0
-1 -1
x = -1
What about if there are negatives?
So the thing about factoring is, the quadratic that you are trying to factor actually gives you all of the information you need.
Whether you need to make a certain number negative or positive, as well as whether or not you need to subtract versus add, you’ll find it in the original quadratic.
You just have to know where to look:
WHAT A QUADRATIC TELLS US:
Let’s take a look at a sample quadratic:
Looking at this quadratic, we can see that the last number is negative:
So, this means that we are going to have alternating signs.
Or, in other words, we’re going to be subtracting.
The reason this works is because the only way to get a negative sign for the last number in our quadratic, is if the two numbers in parenthesis also have different signs.
We can also see that 3 is negative:
Which means the two numbers that are being subtracted, need to end up negative.
So, we’re going to find the factors of 4:
/\
4 1
2 2
And we’re going to subtract them to find what we need.
So:
- = 3
- = 0
As we can see, 4 – 1 = 3, so our factors are going to be:
(x – 4)(x + 1) = 0
Why is 4 negative?
Because we need to make a -3x
x – 4 = 0
x + 1 = 0
+ 4 + 4 -1 -1
x = 4 x = -1
WHAT A QUADRATIC ALSO TELLS US:
Let’s take a look at a sample quadratic:
Looking at this quadratic, we can see that the last number is positive:
So, this means that we are going to have the same sign.
Or, in other words, we’re going to be adding.
This is a little different than the last one.
Because the last number is a positive, this means we will be adding, and both numbers will need to be the same sign.
We can also see that 3 is negative:
Which means the two numbers that are being added, need to end up negative.
So, we’re going to find the factors of 2:
/\
2 1
And we’re going to add them to find what we need.
So:
+ = 3
As we can see, 2 + 1 = 3, so our factors are going to be:
(x – 2)(x - 1) = 0
Why are they both negative?
Because we need to make a -3x
x – 2 = 0
x - 1 = 0
+ 2 + 2 +1 +1
x = 2 x = 1
SO NOW THAT WE HAVE REVIEWED
Let’s look at some examples
Because like a lot of Algebra 2, this is much easier to show than explain:
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