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## Precalculus

# Complex numbers & sum of squares factorization

CCSS.Math:

Learn how expressions of the form x^2+y^2 can be factored into linear factors. This would not be possible without the aid of complex numbers! Created by Sal Khan.

## Video transcript

Voiceover:Long ago in
algebra class, we learned to factor things like X
squared minus Y squared. We saw that this was a
difference of squares that you could factor this
as X plus Y, times X minus Y. This is a little bit of a refresher, you can multiply these two together to verify you get X
squared minus Y squared. In fact, let's just do that for fun. X times X is X squared, X times negative Y is negative XY, Y times X is positive XY, and then Y times negative
Y is minus Y squared. These two middle terms cancel out and your left with X
squared minus Y squared. What I want to tackle in this video is something that we didn't
know how to factor before and that's the sum of the squares. So if we were to factor X,
let me use a brighter color, if we want to try to factor
X squared plus Y squared, before we knew about imaginary numbers, complex numbers, we didn't
know how to factor this. But now that we do,
what I want to try to do and I encourage you
actually to pause the video and try to do it before me,
is to try to express this as a difference of squares,
using the imaginary unit I. So let's try to do this. So the first we want to
write as a difference of squares, so X square
we'll just keep as X squared, but now I want to write
this part, I want to write the second part, right over
here, as subtracting a squares. I want to subtract a square. So we could write this part as subtracting negative Y squared, obviously you subtract a negative that's the
same thing as adding. Now how does this help us? Well this is the same thing as subtracting negative one times Y squared. If we wanted to write this
whole thing as a square, how would we do it? Well we have Y squared and what's negative one the square of? Well we know by definition,
negative one is equal to I squared or that I squared
is equal to negative one. Let's rewrite it that way,
so this is going to be equal to X squared minus,
instead of negative one, I'll write that as I squared,
minus I squared Y squared. All I did was replace this
negative one with an I squared. And now this is interesting. I think you see where this is going, but I'll just make it very explicit. This is now X squared minus IY squared. And just like that, using
I, I've been able to write this sum of squares as
a difference of squares. And now we can factor
just the exact same way we factored this original expression. This thing is going to be equal to X plus IY times X minus IY. And we can verify that
if you multiply these two expressions together, you're going to get X squared plus Y squared. Let's do that, X times X is X squared. X times negative IY is
negative IXY, and then IY times X is positive
IXY, and then finally let me do this in a
color I'm not using yet, finally, IY times negative IY is equal to negative I squared Y squared. Now these middle two terms cancel out, I squared is negative one, and so we have X squared minus negative one Y squared, so this is subtracting a negative, the same thing as adding a positive, and so this simplifies to
X squared plus Y squared. So hopefully this now
gives you an appreciation of using the complex imaginary unit I, how you can actually
factor this expression into essentially the product
of two complex numbers.