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- In this video I'm just going to multiply a ton of
- polynomials, and hopefully that'll give you enough
- exposure to feel confident when you have to multiply any
- for yourselves.
- Let's start with a fairly simple problem.
- Let's say we just want to multiply 2x times 4x minus 5.
- Well, we just straight up use the
- distributive property here.
- And really, when we do all of these polynomial
- multiplications, all we're doing is the distributive
- property repeatedly.
- But let's just do the distributive property here.
- This is 2x times 4x, plus 2x times negative 5.
- Or we could say negative 5 times 2x.
- So you'd say, minus 5 times 2x.
- All I did is distribute the 2x.
- This first term is going to be equal to-- we can multiply the
- coefficients.
- Remember, 2x times 4x is the same thing as-- you can
- rearrange the order of multiplication.
- This is the same thing as 2 times 4, times x times x.
- Which is the same thing as 8 times x squared.
- Remember, x to the 1, times x to the 1, add the exponents.
- I mean, you know x times x is x squared.
- So this first term is going to be 8x squared.
- And the second term, negative 5 times 2 is negative 10x.
- Not too bad.
- Let's do a slightly more involved one.
- Let's say we had 9x to the third power, times 3x squared,
- minus 2x, plus 7.
- So once again, we're just going to do the distributive
- property here.
- So we're going to multiply the 9x to the third times each of
- these terms.
- So 9x to the third times 3x squared.
- I'll write it out this time.
- In the next few, we'll start doing it a
- little bit in our heads.
- So this is going to be 9x to the third times 3x squared.
- And then we're going to have plus-- let me write it this
- way-- minus 2x times 9x to the third, and then plus 7 times
- 9x to the third.
- So sometimes I wrote the 9x to the third first, sometimes we
- wrote it later because I wanted this
- negative sign here.
- But it doesn't make a difference on the order that
- you're multiplying.
- So this first term here is going to be what?
- 9 times 3 is 27 times x to the-- we can add the
- exponents, we learned that in our exponent properties.
- This is x to the fifth power, minus 2 times 9 is 18x to
- the-- we have x to the 1, x to the third-- x
- to the fourth power.
- Plus 7 times 9 is 63x to the third.
- So we end up with this nice little fifth degree
- polynomial.
- Now let's do one where we are multiplying two binomials.
- And I'll show you what I mean in a second.
- This you're going to see very, very, very
- frequently in algebra.
- So let's say you have x minus 3, times x plus 2.
- And I actually want to show you that all we're doing here
- is the distributive property.
- So let me write it like this: times x plus 2.
- So let's just pretend that this is one big number here.
- And it is.
- You know, if you had x's, this would be some number here.
- So let's just distribute this onto each of these variables.
- So this is going to be x minus 3, times that green x, plus x
- minus 3, times that green 2.
- All we did is distribute the x minus 3.
- This is just the distributive property.
- Remember, if I had a times x plus 2, what would
- this be equal to?
- This would be equal to a times x plus a times 2.
- So over here, you can see when x minus 3 is the same thing as
- a, we're just distributing it.
- And now we would do the distributive property again.
- In this case, we're distributing the x now onto
- the x minus 3.
- We're going to distribute the 2 onto the x minus 3.
- You might be used to seeing the x on the other side, but
- either way, we're just multiplying it.
- So this is going to be-- I'll stay color coded.
- This is going to be x times x, minus 3 times x, plus x times
- 2-- I'm going through great pains to keep it
- color coded for you.
- I think it's helping-- minus 3 times 2.
- All I did is distribute the x and distribute the 2.
- And soon you're going to get used to this.
- We can do it in one step.
- You're actually multiplying every term in this one by
- every term in that one, and we'll figure out faster ways
- to do it in the future.
- But I really want to show you the idea here.
- So what's this going to equal?
- This is going to equal x squared.
- This right here is going to be minus 3x.
- This is going to be plus 2x.
- And then this right here is going to be minus 6.
- And so this is going to be x squared minus 3 of something,
- plus 2 of something, that's minus 1 of that something.
- Minus x, minus 6.
- We've multiplied those two.
- Now before we move on and do another problem, I want to
- show you that you can kind of do this in your head as well.
- You don't have to go through all of these steps.
- I just want to show you really that this is just the
- distributive property.
- The fast way of doing it, if you had x minus 3, times x
- plus 2, you literally just want to multiply every term
- here times each of these terms.
- So you'd say, this x times that x, so
- you'd have x squared.
- Then you'd have this x times that 2, so plus 2x.
- Then you'd have this minus 3 times that x, minus 3x.
- And then you have the minus 3, or the negative 3, times 2,
- which is negative 6.
- And so when you simplify, once again you get x squared
- minus x minus 6.
- And it takes a little bit of practice to
- really get used to it.
- Now the next thing I want to do-- and the principal is
- really the exact same way-- but I'm going to multiply a
- binomial times a trinomial, which
- many people find daunting.
- But we're going to see, if you just stay calm,
- it's not too bad.
- 3x plus 2, times 9x squared, minus 6x plus 4.
- Now you could do it the exact same way that we did the
- previous video.
- We could literally take this 3x plus 2, distribute it onto
- each of these three terms, multiply 3x plus 2 times each
- of these terms, and then you're going to distribute
- each of those terms into 3x plus 2.
- It would take a long time and in reality, you'll never do it
- quite that way.
- But you will get the same answer we're going to get.
- When you have larger polynomials, the easiest way I
- can think of to multiply, is kind of how you
- multiply long numbers.
- So we'll write it like this.
- 9x squared, minus 6, plus 4.
- And we're going to multiply that times 3x plus 2.
- And what I imagine is, when you multiply regular numbers,
- you have your ones' place, your tens' place, your
- hundreds' place.
- Here, you're going to have your constants' place, your
- first degree place, your second degree place, your
- third degree place, if there is one.
- And actually there will be in this video.
- So you just have to put things in their proper place.
- So let's do that.
- So you start here, you multiply almost exactly like
- you would do traditional multiplication.
- 2 times 4 is 8.
- It goes into the ones', or the constants' place.
- 2 times negative 6x is negative 12x.
- And we'll put a plus there.
- That was a plus 8.
- 2 times 9x squared is 18x squared, so we'll put that in
- the x squared place.
- Now let's do the 3x part.
- I'll do that in magenta, so you see how it's different.
- 3x times 4 is 12x, positive 12x.
- 3x times negative 6x, what is that?
- The x times the x is x squared, so it's
- going to go over here.
- And 3 times negative 6 is negative 18.
- And then finally 3x times 9x squared, the x times the x
- squared is x to the third power.
- 3 times 9 is 27.
- I wrote it in the x third place.
- And once again, you just want to add the like
- terms. So you get 8.
- There's no other constant terms, so it's just 8.
- Negative 12x plus 12x, these cancel out.
- 18x squared minus 18x squared cancel out, so we're just left
- over here with 27x to the third.
- So this is equal to 27x to the third plus 8.
- And we are done.
- And you can use this technique to multiply a trinomial times
- a binomial, a trinomial times a trinomial, or really, you
- know, you could have five terms up here.
- A fifth degree times a fifth degree.
- This will always work as long as you keep things in their
- proper degree place.