I want to combine these two nasty rational expressions. Before I can do that, I need to see if they have the same denominator or not. Looking at it, the first thing that comes to mind right away is that those don't look the same. So in order to tell how similar they are, or what factors they have in common, I need to factor each of those.
I notice that that's the difference of perfect squares. So I can factor easily just using the patterns that I know. This guy I'm going to factor as x plus 4, x plus 4. So let's talk about what this means. Both of these denominator have an x plus 4 piece. This guy here has two x plus 4 pieces which means this original fraction needs another x plus 4. I've got to multiply by x plus 4 over x plus 4 on that first guy so that it will have two x plus 4s in the bottom.
Looking at the second fraction, he has two x plus 4s, but he's missing the x minus 4. That means this fraction needs to get multiplied by x minus 4 over x minus 4. It's really tricky, because I have three factors in my denominator. Well, let's talk about what's going to happen next. My denominator is going to look like all three of those things multiplied together. X minus 4, x plus 4, and x plus 4 again.
On top I'm going to have x times x plus 4 for my first fraction. That's my first fraction, subtract second fraction's top; x times x minus 4. Tons of 4s. Tons of pluses, and minus. Tons of x's. I know it's confusing guys that's why it's really important you show all your work.
To continue on I'm going to go on and distribute, and combine like terms. So let me show you what distributing looks like. Before I do that, I'm going to tell you something that's going on in my brain. When I'm writing out these problems, I leave lots of space, because I do go back and write in little things in between the lines.
So, if you're doing your homework, you might want to do this on blank paper maybe, or if you're using line paper, that's fine. Maybe skip some lines, so you can go back and write in these little distributing things like you're doing.
So here is what I did. I multiplied this first guy x² plus 4x. Then I'm subtracting x², and then minus minus, which is the same as plus 4x. Let's combine the numerators. So I'll have x² take away x² is no x². 4x plus 4x is 8x on top.
Then on bottom, I just have that whole big product of three binomials. I can write it like that x plus 4². That's my final most simplified form. Don't be tempted to cancel out the 8 with those 4's, because those 4's are attached to the x's. You can't break apart that x minus 4 difference. Those guys are like a unit. You can't cancel anything out.
The last thing I'm going to do is write my excluded solutions. The excluded value is any value of x that would make this denominator equal to 0. So the denominator would be equal to 0 if x was equal to 4, or -4. So I'm going to write those as my excluded values.
So guys the last thing I want to leave you with is, when you're doing these homework problems leave lots of space. They take up lots of lines. Maybe use online paper it's up to you. Then also make sure you have the exact same denominators in both of your fractions. It helps to factor them first, so you can see which factor each of the fraction is missing.
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M.A. in Secondary Mathematics, Stanford University B.S., Stanford University
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