Adding Rational Expressions Example 3

Let's do one more fairly involved example of adding, or in this case actually subtracting rational expressions. Let's say I have 4 over 9x squared minus 49. And from that I want to subtract-- maybe I'll do this in different color-- I want to subtract 1 over 3x squared plus 5x, minus 28. So just like we saw in the last video, we want to find the least common multiple of these two expressions. And the way to figure out that is to factor these two things. Now this first expression, you might recognize, is a difference of squares. This is 3x squared minus 7 squared. So we can rewrite it as 4 over 3x-- well, let me write it this way-- 3x plus 7, times 3x minus 7. 3x squared is 9x squared. 7 squared is 49. Difference of squares. And from that we're going to subtract this thing over here. Let's see how we can factor that out. Well, once again, this has a non-1 coefficient here, so you can always use the Pythagorean theorem, but that actually gets messy if you actually just want to factor as opposed to getting roots. So the best thing here is to do grouping. So let me factor it over here. I need to find two numbers. Let me just write it over here. So we have 3x-- let me write it. I want to write it someplace where I can-- well, actually I'll write it here. So I have 3x squared plus 5x, minus 28. I need to find two numbers that when I multiply them I get 3 times negative 28. So it's negative 60 and negative 24, that's negative 84, if I multiply those two numbers. And a plus b should be equal to 5. So what numbers? Well, it's 7 and 12. Your times table. 7 times 12 is 84. So if you make this a 12 and you make this a negative 7, then it works. 12 minus 7 is 5, 12 times negative 7 is 84. So I can rewrite this as 3x squared. I'll group the 12 with the 3, because they have a common factor of 3. So plus 12x minus 7x, minus 28. Once again, 7 and 28 have a common factor, so it's looking good. And just to reiterate, all I did is I rewrote this 5x as 12x minus 7x. These are, obviously, the same thing. 12x minus 7x is 5x. But then we can group them just like that. If we factor out a 3x here, this first expression is the same thing as 3x times x plus 4. And in the second expression over here-- I'll do it in red, or in white I guess-- let's factor out a negative 7. So minus 7 times x plus 4. And lucky for us, we have a common factor is x plus 4, so we can factor that out. So if you factor out the x plus 4, you get x plus 4 times 3x minus 7. So that's what this guy over here will factor into. So I can rewrite it right here. So this expression is the exact same thing as 1 over-- well, I can write the 3x minus 7 first-- 3x minus 7 times x plus 4. This and this is the exact same statement. So let's figure out what our common denominator is here. So our common denominator-- I'll write it like that-- it needs to have all of the factors of this guy right here, so it needs to have a 3x plus 7. It needs to have a 3x minus 7. And also it has to have all of the factors of this guy. Well, it already has a 3x minus 7, so this is already done. But it doesn't have an x plus 4 in it yet. So now I have to multiply it by x plus 4. So clearly, if you look at this, this is clearly divisible by this guy, because it has all of its factors. You could divide this out. And it's clearly divisible by this guy. You have 3x minus 7 times x plus 4. 3x minus 7 times x plus 4. So this is the least common multiple of these two expressions. Now, how do we get-- so what do we have to multiply this guy by to have this as the denominator? Well, I just have to multiply the numerator and the denominator by x plus 4. So if I multiply this guy-- let me scroll to the left a little bit-- by x plus 4 over x plus 4, my denominator will work out. And so my numerator will become 4 times x plus 4, which is 4x plus 16. That's this first term over here. And then what do I have to multiply this guy by to have the same denominator as this? Well, I have a 3x minus 7 times x plus 4. 3x minus 7 times x plus 4. I have to multiply it by 3x plus 7. So I also have to multiply the numerator by 3x plus 7. So this is 4x plus 16 minus 1 times 3x plus 7, or minus 3x plus 7 is the same thing as minus 3x minus 7. I'm just distributing the negative sign. And then what is this going to be equal to? The numerator, we have a 4x minus a 3x will just be an x. And then we have a 16 minus a 7, which is a 9. And then the denominator, we, of course, have all of this stuff over here, which is 3x plus 7, times 3x minus 7, times x plus 4. And we're done.