Factoring polynomials using complex numbers | Khan Academy

We're told that Ahmat tried to write ( x^4 + 5x^2 + 4 ) as a product of linear factors. This is his work, and then they tell us all the steps that he did, and then they say in what step did Ahmad 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^4 + 10x^2 + 9 ), and it looks like Ahmad tried to factor that into ( (x^2 + 9)(x^2 + 1) ). And this indeed does make sense because if we said that let's say ( u ) is equal to ( x^2 ), we could rewrite this right over here as ( u^2 + 10u + 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.

Then we've learned how to factor things like this many times. We look, and we say, "Okay, what two numbers when I add them I get 10, and when I multiply them I get 9?" It would be 9 and 1. So you could write this as ( (u + 9)(u + 1) ). And of course, if ( u ) is equal to ( x^2 ), this would be ( (x^2 + 9)(x^2 + 1) ), which is exactly what Ahmad has right over here. So step 1 is looking great.

All right, now let's think about what Ahmad did in step two. They didn't do anything to ( x^2 + 9 ), but it looks like they tried to further factor ( x^2 + 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 + a)(x - a) ).

We could have a sum of squares if we're thinking about complex numbers; this is going to be ( (x + ai)(x - ai) ). And in this situation, well, the ( x ) is ( x ), and then our ( a ) would be 1. So we're going to have ( (x + i)(x - i) ). So step 2 is looking great, and now let's go to step three.

So in step three, there’s no change to this part of the expression, and it looks like Ahmad is trying to factor ( x^2 + 9 ) based on the same principle. Now, ( x^2 + 9 ) is the same thing as ( x^2 + 3^2 ). So if you use this exact same idea here, if you factor it, it should be ( (x + 3i)(x - 3i) ).

But what we see over here is Ahmad took the square root of three instead of just having a three here. Ahmad 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 Ahmad makes his first mistake, and we're done.