Factoring polynomials using complex numbers

- [Instructor] We're told that Amat tried

to write x to the fourth
plus 5x squared plus 4

as a product of linear factors.

This is his work.

And then they'd tell us all
of the steps that he did.

And then they say, in what step

did Amat make his first mistake?

So pause this video and see
if you can figure that out.

All right, now let's work
through this together.

So we're starting with x to
the fourth plus 10x squared

plus 9.

And it looks like Amat
tried to factor that

into x squared plus 9
times x squared plus 1.

And this indeed does make sense,

because if we said that let's say,

u is equal to x squared,

we could rewrite this right over here

as u squared plus 10u plus 9.

The whole reason why you would do this is

so that you could write
this higher order expression

in terms of a second degree expression.

And then we've learned
how to factor things like

this many times.

We look, we say, okay,

what two numbers when I add them I get 10,

and when I multiply them I get nine,

and it would be nine and one?

And so you could write this

as u plus 9 times u plus 1.

And of course, if u is equal to x squared,

this would be x squared plus
9 times x squared plus 1.

Which is exactly what
Amat has right over here.

So step one is looking great.

All right, now let's think
about what Amat did in step two.

They didn't do anything
to x squared plus 9

but it looks like they tried

to further factor x squared plus 1.

And this does seem right.

We just have to remind ourselves just

as you have a difference of squares

if you're dealing with
non-complex numbers,

so we could rewrite this right over here

as x plus a times x minus a.

We could have a sum of squares

if we're thinking about complex numbers.

This is going to be x
plus ai times x minus ai.

And in this situation while the x is x

and then our a would be one.

So we're going to have x plus i,

x plus i times x minus i .

So step two is looking great.

And now let's go do step three.

So in step three,

no change to this part of the expression.

And it looks like Amat is trying

to factor x squared plus 9
based on the same principle.

Now x squared plus 9 is the same thing

as x squared plus 3 squared.

So if you use this exact same idea here,

if you factored it should be
x plus 3i times x minus 3i.

But what we see over here

is Amat took the square root of three,

instead of just having a three here.

Amat treated it instead
of having a nine here

as if we actually had a three

so they made a little
bit of an error there.

So this is the step where
Amat makes his first mistake

and we're done.