All right in this video. I want to talk about factoring sums and differences of cubes, and there's two formulas
You've got to know
the first one says if you have something cubed minus something cubed you get the quantity so x cubed minus y cubed we get the
Quantity x minus y
Times the quantity x squared plus xy plus y squared [it]
Says if we have something cubed plus something cubed so x cubed plus y cubed we get the quantity x plus y?
Times the quantity x squared minus xy plus y squared a couple remarks
notice the formulas are the same except for
the
the negative and the positive switch out in the first two spots, so
Starts negative positive, and then positive negative, okay assuming. You write the binomial term first
also
recall that if you're doing differences of
perfect squares that factors, but not sums so remember if they work if there were
Exponents of two up here there was no formula that goes with x squared plus y squared
But when [they're] cubes you can factor both of them
So just one little thing I think worth remembering so let's do a couple examples, so the first one
I'm going to do here is 64 x cubed plus 125 and then we've got
27 y cubed minus 8
So let me see if I can bring my formula back in here real quick
Okay, so it looks like in this one. We'll have to use the bottom formula
Okay, so it looks like we have something q plus something cubed
But the first thing I'm going to do is I'm going to just rewrite it a little bit, okay
So I'm going to rewrite my 64 x cubed and what I'm going to do is I want
I want to write that as something cubed
Plus something else cubed, okay?
[that's] kind of how I'm interpreting this force first or the second formula something cubed plus something cubed
Well, I guess to get 64 we would need 4 cubed and to get x cubed. We just need to stick an x in there
Likewise to get [125] [we] would need 5 cubed
So now I'm going to use this formula. So instead of just x + y I've got 4 x and I've got a positive 5
so instead of x I've got 4 x and then it just says write plus, whatever is in the second set of parentheses and
Then it says whatever was in the first set of parentheses
okay, it says now take that thing and
Square it, so I'm taking my 4 x and I'm squaring it then it says write a minus sign
It says take the first thing times the second one so [4] x times 5
And then it says write a plus, and it says whatever number was being originally cubed
it says now we take that number and we
Square it, so
[certainly] a little you know. I think it takes a little getting used to for sure
[if] we simplify this we have 4x plus 5 [in] the front not much to do there
And I'm going to take 4 x and square it [so] 4x times 4x would be
16x Squared
we have negative 4x times 5 well I can multiply the negative 4 and the 5 and make that negative 20x and
Then our positive 5 Squared will just get positive 25
Okay
So one more here. I'm going to do the same thing
I have something cubed minus something well okay, so the trick is we have to again kind of rewrite them both
So I think what number cubed would give me 27 and there?
We would just need a three and to get the y cubed we would need a y in the parentheses
[minus] and now I think what number cubed is eight?
Excuse me. I wrote it, [too]
I'm getting ahead of myself, so I think two cubed is going to give us positive eight and
Now we just do the same thing. We [just] use our little formula here
Okay, so don't get confused by the x and the y here and the fact that we have y's and numbers to me
We're using this formula, and it says if you have something cubed minus something cubed
it says whatever was being cubed so kind of whatever's downstairs write that first, so we'll get a 3y
Then it says write a minus sign, and it says whatever was a also being cubed
[it] says just stick that next to it. So we'll [just] drop our positive two down and
Then it says take the first thing in the first set of parentheses and square it
Then write a plus and now multiply the things inside the parenthesis together
so 3y times 2 and then
[it] says whatever number was out there. It says we just square that as well, okay?
[so] if we simplify this we'll get 3y
minus 2
3y times 3y will give us
9y squared
3y times 2 will give us positive 6y and
Then 2 Squared will give us a positive 4 and again we have now factored
27y Cubed minus 8
[ok] so again. I think these are formulas that you don't run into all the [time]
You'll see them a little bit now and again. You know in calculus, or you know maybe in a precalculus class
There's certainly formulas that people forget
so
even if you forget it at least know that there is a formula out there and
That way if you're doing a homework problem or something at least you'll know hey
You know should dig out my algebra book and look it up so again
You know just a little practice, but again nothing too crazy just a few more formulas that
Unfortunately, you got to know