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- In this video, we're going to learn to divide polynomials,
- and sometimes this is called algebraic long division.
- But you'll see what I'm talking about
- when we do a few examples.
- Let's say I just want to divide 2x plus 4 and
- divide it by 2.
- We're not really changing the value.
- We're just changing how we're going to express the value.
- So we already know how to simplify this.
- We've done this in the past. We could divide the numerator
- and the denominator by 2, and then this
- would be equal to what?
- This would be equal to x plus 2-- let me write it this way--
- it would be equal to, if you divide this by 2,
- it becomes an x.
- You divide the 4 by 2, it becomes a 2.
- If you divide the 2 by 2, you get a 1.
- So this is equal to x plus 2, which is pretty
- straightforward, I think.
- The other way is you could have factored a 2 out of here,
- and then those would have canceled out.
- But I'll also show you how to do it using algebraic long
- division, which is a bit of overkill for this problem.
- But I just want to show you that it's not fundamentally
- anything new.
- It's just a different way of doing things, but it's useful
- for more complicated problems.
- So you could have also written this as 2 goes into 2x plus 4
- how many times?
- And you would perform this the same way you would do
- traditional long division.
- You'd say 2-- you always start with the highest degree term.
- 2 goes into the highest degree term.
- You would ignore the 4.
- 2 goes into 2x how many times?
- Well, it goes into 2x x times and you put
- the x in the x place.
- x times 2 is 2x.
- And just like traditional long division, you now subtract.
- So 2x plus 4 minus 2x is what?
- It's 4, right?
- And then 2 goes into 4 how many times?
- It goes into it two times, a positive two times.
- Put that in the constants place.
- 2 times 2 is 4.
- You subtract, remainder 0.
- So this might seem overkill for what was probably a
- problem that you already knew how to do and
- do it in a few steps.
- We're now going to see that this is a very
- generalizable process.
- You can do this really for any degree polynomial dividing
- into any other degree polynomial.
- Let me show you what I'm talking about.
- So let's say we wanted to divide x plus 1 into x squared
- plus 3x plus 6.
- So what do we do here?
- So you look at the highest degree term here, which is an
- x, and you look at the highest degree term here,
- which is an x squared.
- So you can ignore everything else.
- And that really simplifies the process.
- You say x goes into x squared how many times?
- Well, x squared divided by x is just x, right? x goes into
- x squared x times.
- You put it in the x place.
- This is the x place right here or the x to
- the first power place.
- So x times x plus 1 is what?
- x times x is x squared.
- x times 1 is x, so it's x squared plus x.
- And just like we did over here, we now subtract.
- And what do we get?
- x squared plus 3x plus 6 minus x squared-- let me be very
- careful-- this is minus x squared plus x.
- I want to make sure that negative sign only-- it
- applies to this whole thing.
- So x squared minus x squared, those cancel out.
- 3x, this is going to be a minus x.
- Let me put that sign there.
- So this is minus x squared minus x, just to be clear.
- We're subtracting the whole thing.
- 3x minus x is 2x.
- And then you bring down the 6, or 6 minus 0 is nothing.
- So 2x plus 6.
- Now, you look at the highest degree term, an x and a 2x.
- How many times does x go into 2x?
- It goes into it two times.
- 2 times x is 2x.
- 2 times 1 is 2.
- So we get 2 times x plus 1 is 2x plus 2.
- But we're going to want to subtract this from this up
- here, so we're going to subtract it.
- Instead of writing 2x plus 2, we could just write negative
- 2x minus 2 and then add them.
- These guys cancel out.
- 6 minus 2 is 4.
- And how many times does x go into 4?
- We could just say that's zero times, or we could say that 4
- is the remainder.
- So if we wanted to rewrite x squared plus 3x plus 6 over x
- plus 1-- notice, this is the same thing as x squared plus
- 3x plus 6 divided by x plus 1, this thing divided by this, we
- can now say that this is equal to x plus 2.
- it is equal to x plus 2 plus the remainder divided by x
- plus 1 plus 4 over x plus 1.
- This right here and this right here are equivalent.
- And if you wanted to check that, if you wanted to go from
- this back to that, what you could do is multiply this by x
- plus 1 over x plus 1 and it add the two.
- So this is the same thing as x plus 2.
- And I'm just going to multiply that times x plus
- 1 over x plus 1.
- That's just multiplying it by 1.
- And then to that, add 4 over x plus 1.
- I did that so I have the same common denominator.
- And when you perform this addition right here, when you
- multiply these two binomials and then add the 4 up here,
- you should you get x squared plus 3x plus 6.
- Let's do another one of these.
- They're kind of fun.
- So let's say that we have-- we want to simplify x squared
- plus 5x plus 4 over x plus 4.
- So once again, we can do our algebraic long division.
- We can divide x plus 4 into x squared plus 5x plus 4.
- And once again, same exact process.
- Look at the highest degree terms in both of them. x goes
- into x squared how many times?
- It goes into it x times.
- Put it in the x place.
- This is our x place right here.
- X times x is x squared.
- x times 4 is 4x.
- And then, of course, we're going to want to subtract
- these from there.
- So let me just put a negative sign there.
- And then these cancel out.
- 5x minus 4x is x.
- 4 minus 0 is plus 4.
- x plus 4, and then you could even see this coming.
- You could say x plus 4 goes into x plus 4 obviously one
- time, or if you were not looking at the constant terms,
- you would completely just say, well, x goes
- into x how many times?
- Well, one time.
- Plus 1.
- 1 times x is x.
- 1 times 4 is 4.
- We're going to subtract them from up here, so it cancels
- out, so we have no remainder.
- So this right here simplifies to-- this is
- equal to x plus 1.
- And there's other ways you could have done this.
- We could have tried to factor this numerator.
- x squared plus 5x plus 4 over x plus 4.
- This is the same thing as what?
- We could have factored this numerator as x plus 4
- times x plus 1.
- 4 times 1 is 4.
- 4 plus 1 is 5, all of that over x plus 4.
- That cancels out and you're left just with x plus 1.
- Either way would have worked, but the algebraic long
- division will always work, even if you can't cancel out
- factors like that, even if you did have a remainder.
- In this situation, you didn't.
- So this was equal to x plus 1.
- Let's do another one of these just to make sure that you
- really-- because this is actually a very, very useful
- skill to have in your toolkit.
- So let's say we have x squared-- let me
- just change it up.
- Let's say we had 2x squared-- I could really make these
- numbers up on the fly.
- 2x squared minus 20x plus 12 divided by-- actually, let's
- make it really interesting, just to show you that it'll
- always work.
- I want to go above quadratic.
- So let's say we have 3x to the third minus 2x squared plus 7x
- minus 4, and we want to divide that by x squared plus 1.
- I just made this up.
- But we can just do the algebraic long division to
- figure out what this is going to be or what this
- simplified will be.
- x squared plus 1 divided into this thing up here, 3x to the
- third minus 2x squared plus 7x minus 4.
- Once again, look at the highest degree term.
- x squared goes into 3x to the third how many times?
- Well, it's going to go into it 3x times.
- You multiply 3x times this, you get 3x to the third.
- So it's going to go into it 3x times.
- So you have to write the 3x over here in the x terms. So
- it's going to go into it 3x times, just like that.
- Now let's multiply.
- 3x times x squared is 3x to the third, right?
- 3x times squared plus 3x times 1.
- So we have a 3x over here.
- I'm making sure to put it in the x place.
- And we're going to want to subtract them.
- And what do we have?
- What do we have when we do that?
- These cancel out.
- We have a minus 2x squared.
- And then 7x minus-- because I just subtracted 0 from there--
- 7x minus 3x is plus 4x, and we have a minus 4.
- Once again, look at the highest degree term.
- x squared and a negative 2x squared.
- So x squared goes into negative 2x
- squared negative 2 times.
- Negative 2, put it in the constants place.
- Negative 2 times x squared is negative 2x squared.
- Negative 2 times 1 is negative 2.
- Now, we're going to want to subtract these from there, so
- let's multiply them by negative 1, or
- those become a positive.
- These two guys cancel out.
- 4x minus 0 is-- let me switch colors-- 4x minus 0 is 4x.
- Negative 4 minus negative 2 or negative 4 plus 2 is equal to
- negative 2.
- And then x squared, now it has a higher degree than 4x, and
- the highest degree here, so we view this as the remainder.
- So this expression we could rewrite it as being equal to
- 3x minus 2-- that's the 3x minus 2-- plus our remainder
- 4x minus 2, all of that over x squared plus 1.
- Hopefully, you found that as fun as I did.