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- 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.