U1S4V13 Factoring the difference of nth powers (optional).txt

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In this video, we're going to talk
about how to factor a difference of nth powers.
It's pretty neat.
You've probably learned how to factor a difference of squares.
So you get b minus x times b plus x.
And some of you may even have seen
how to factor a difference of cubes.
You're going to get this b minus x factor again,
and then you get b squared plus bx plus x squared
as the second factor.

For the difference of nth, powers
we're just going to continue this pattern.
So I claim that b to the n minus x to the n
will factor as we're going to have this b minus x term.
And then as the second factor, we'll have b to the n minus 1
plus b to the n minus 2 times x plus b to the n minus 3 times
x squared plus et cetera, et cetera, plus b times x
to the n minus 2.
And then finally, x to the n minus 1.

So let's just multiply this out and see that we do,
in fact, get b to the n minus x to the n.
So I'll do the terms involving this b first.
When we multiply all of this stuff by b,
we're going to get b to the n plus b to the n minus 1 times
x plus b to the n minus 2 times x
squared plus et cetera, et cetera, plus b squared times x
to the n minus 2.
And then finally, b times x to the n minus 1.
So that's everything here multiplied by the b,
and then we have this minus x that we still
need to multiply by everything.
And we're going to get minus b to the n minus 1 times x
minus b to the n minus 2 times x squared, et cetera, et cetera,
et cetera.
The next-to-last term is minus b times x to the n minus 1.
And then finally, minus x to the n.
So we've multiplied everything out, and look what happens.
All of this stuff cancels.
So this cancels this.
This cancels this.
These cancel over here.
The only things that don't cancel
are this b to the n and this minus x to the n.
So this whole thing just equals b to the n minus x to the n,
as I promised.