Dividing polynomials
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Polynomial Division
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Polynomial divided by monomial
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Dividing multivariable polynomial with monomial
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Dividing polynomials 1
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Dividing polynomials with remainders
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Synthetic Division
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Synthetic Division Example 2
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Why Synthetic Division Works
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Factoring Sum of Cubes
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Difference of Cubes Factoring
Algebraic Long Division Dividing one polynomial into another
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- I've been asked to make a video on algebraic division or
- algebraic long division.
- So I'll make a video an algebraic long division.
- I'm just going to make up a problem.
- Let's say we wanted to divide-- we wanted to see how many times
- does-- I'll start with a fairly straightforward problem.
- How many times does 2x plus 1 go into-- I don't know-- let's
- say it's 8x to the third minus 7x squared plus 10x minus 5.
- So what we do is we just take-- actually, just the exact same
- way that you would do with long division, traditional long
- division of multiple digits.
- In the 2x plus 1 expression you look at, oh, what is
- the highest degree term?
- And that's really all we're going to pay attention
- to most of the time.
- So the first step is you say, OK, the highest
- degree term is 2x.
- How many times does 2x go into the highest degree term of the
- number-- not the number-- the expression that we're
- dividing into?
- So you say, how many times does 2x go into 8x to the third?
- Well, we could do a little division on the side, but you
- could imagine eventually this is pretty straightforward.
- So if you have 8x to the third divided by 2x, that
- is equal to 4x squared.
- So 2x goes into 8x to the third 4x squared times.
- And this is the key thing.
- You don't want to write the 4x squared here.
- You want to keep everything in the correct places.
- So when you're dividing numbers you think of the ones, the
- tens, the hundreds, and the thousands place et cetera.
- When you're dividing polynomials you can kind of
- think of the x to the 0 space.
- The x to the 1 space or the x space.
- The x squared space.
- The x to the third space.
- So, when we say that 2x goes into 8x to the third 4x squared
- times let's write that in the x squared spot.
- It goes into it 4x squared times.
- Now, we take that 4x squared and we multiply
- it by our expression.
- I think you're already seeing that this is very similar
- to long division.
- And actually, if x was a ten, it would be identical
- to long division.
- And I'll let you think about that.
- If x was 10 this would be the thousands place.
- This would be 8,000 minus-- although you would have
- negative digits, which doesn't make a bunch of sense.
- But I think you get what I'm saying.
- But anyway, back to this algebraic long division.
- Although I think it is very important to see the
- parallels between this and traditional long division.
- Well, anyway, we said that 2x goes into 8x to the
- third 4x squared times.
- Now what we can do is we can multiply 4x squared
- times 2x plus 1.
- So 4x squared times 1, that's 4x squared.
- So we can write that in the x squared's place.
- We could write it 4x squared.
- And 4x squared times 2x is 8x to the third.
- This is plus here.
- And now, just like we do with traditional long division, we
- can subtract this from this.
- So minus 7x squared minus 4x squared is minus 11x squared.
- And then 8x to the third minus 8x to the third is 0, so we
- can ignore that right there.
- And if we want, we can bring down the rest of the number,
- but maybe just for fun we'll bring down the next spot just
- like we do in traditional long division.
- Actually, let me erase this over here.
- Because I think we might find that real estate useful.
- All right, I'm back.
- Actually, it doesn't hurt to bring down the whole thing.
- Just so that you understand what we're doing.
- We're saying if you were to divide 2x plus 1 to this entire
- expression, and you say it goes in 4x squared times.
- Now you can kind of call it our intermediate
- remainder is what's left.
- This is what's left.
- You could almost imagine 4x squared times 2x plus 1 is--
- this is 8x to the third plus 4x squared plus 0 plus 0 because
- it doesn't contribute any thing to these spots.
- But then what's left over is this expression.
- If you take this minus this whole expression, you
- get what's left over.
- Now we just do the same thing over.
- How many times does 2x-- we just look at the
- highest order term.
- How many times does 2x go into negative 11x squared?
- So let's write it here on the side again.
- Actually, let me do it here.
- So if we were to take minus 11x squared divided by 2x,
- that is equal to what?
- That is equal to minus 11/2 x.
- So 2x goes into minus 11x squared minus 11/2 x times.
- So we'll write that in our x place.
- So minus 11/2.
- We could write that as 5.5.
- I'll just write it as a fraction.
- Minus 11/2 x.
- And now, what is minus 11/2 x times 2x plus 1?
- So minus 11/2 x times 1 is minus 11/2 x.
- And we'll want to write that in the x position.
- I'll switch colors just to not be monotonous.
- So minus 11/2 x times 1 is minus 11/2 x.
- And then, minus 11/2 x times 2x, well, we
- should know that is.
- But you can multiply them out.
- It'll be minus 11x squared.
- I think you see what we're doing.
- After every step we're canceling out the largest
- degree of the polynomial we're dividing into.
- Fair enough?
- Now let's subtract this expression from this.
- And we'll get kind of our new intermediary remainder.
- And maybe that'll be the full remainder.
- So let's see.
- Minus 11x squared minus 11x squared.
- That's 0, so we don't have to write anything there.
- 10x minus negative 11/2 x.
- Remember, we're subtracting this negative number from 10x.
- So if you're subtracting a negative number it's like
- adding a positive number.
- So you could view this as 10 plus 11/2.
- So 10 plus 11/2, that's 20/2 plus 11/2.
- That's 31/2 or 15.5.
- I'll just write 31/2 x.
- 31/2 x.
- And then you could say that there was a 0 here and
- when you subtract 0 from minus 5 you get minus 5.
- And now we say, how many times does 2x go into 31/2 x.
- Let's do a little work on the side here.
- So if I have 31/2 x divided by 2x.
- Well, the x's will just cancel out.
- This is equal to 31/4.
- This is the same thing as 31 over 2 times 1/2.
- So it's 31/4.
- So 2x goes into this expression 31/4 times and I'll
- switch colors.
- I'll switch to green.
- And that's a positive, right?
- You're dividing a positive into a positive.
- So plus 31/4 times.
- And I'm writing that in the-- you could view that in the
- constant space or the x to the 0 space.
- Or the 1 space even.
- So it goes into it 31/4 times.
- 31/4 times 1 is 31/4.
- And 31/4 times 2x is 31/2 x.
- And now we subtract.
- This is a plus here.
- We subtract the green expression from the light
- blue expression and we're left with this.
- When you subtract this from this you're left with 0, so
- nothing shows up there.
- And we're left with minus 5 minus 31/4.
- And we can just do a little bit of fraction work here.
- That's equal to, let's see.
- Minus 5 over 4 is minus 20 minus 31.
- All of that over 4.
- So that is equal to what?
- Minus 20, that's equal to minus 51.
- Minus 51/4.
- So, our answer is 2x plus 1 goes into 8x to the third minus
- 7x squared plus 10x minus 5-- it goes into it 4x squared
- minus 11/2 x plus 31/4 times.
- But there is a remainder, and this is the remainder.
- And so a way to visualize this or another way to think about
- this problem so it's actually useful when we're actually
- solving real problems.
- And that you just don't view this as some kind of mechanical
- way to get problems right on a test that only tests
- algebraic long division.
- As another way to write this relationship you could write
- that-- let me do it in another color.
- I've used many of my colors already.
- So you can write that 2x plus 1 times this-- 4x squared.
- That's an x.
- Minus 11/2 x plus 31/4.
- Plus the remainder.
- So when you multiply these two out, and then if you were to
- add the remainder-- 51/4-- that that would equal-- and let
- me draw a dividing line.
- I don't want to confuse you with all this stuff here.
- That would equal this.
- That would equal 8x to the third minus 7x squared
- plus 10x minus 5.
- Anyway, I hope that helps.
- See you in the next video.
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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