yego.me
💡 Stop wasting time. Read Youtube instead of watch. Download Chrome Extension

Factoring polynomials using complex numbers | Khan Academy


2m read
·Nov 10, 2024

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.

More Articles

View All
15 Things You Say You'll Do But Don't
Sometimes it seems like we enjoy letting ourselves down. When we look at all the things we say we’ll do but don’t do, it seems crazy. Many of those things are totally incompatible with your skills, lifestyle, and history of sticking to goals. And funny en…
Fired Up About Dark Matter | StarTalk
All right, number two. This next question is from, okay, let’s see. This is, uh, this is from David Crosby. Oh, okay, and in his interview with you, he asked me, he was asking me questions. You tell me, you snap, you clipped the question. I clipped a que…
Thoughts on the nation's report card
Hi folks, Sal here from Khan Academy. Many of you all have caught wind that the National Assessment of Educational Progress just came out, also known as the NAEP or the Nation’s Report Card, and the results were not good. They were already bad pre-pandemi…
WE ATE GOAT BRAINS - Smarter Every Day 20
(African music) (Destin) Okay, Smarter Every Day. Pringles can will make the absolute perfect spaghetti holder when you’re done with the Pringles. There you go. Reuse. Recycle, reduce, reuse, in Africa. Right? Hey, Bob Marley-pants, are you making spa…
WHY IS THERE A MOON? .... and more!
Hey, Vsauce. Michael here. And I’m at Regent’s Park in London, joined by Tom from “/Tom.” Awesome channel, check it out. This camera belongs to Tom, and Hazel is operating it right now. Yeah, that’s her giving a thumbs up, which I’m sure just looks like a…
Making Custard | Live Free or Die: How to Homestead
[Music] Custard utilizes ingredients that we tend to have a lot of, so I want to teach you how to make custard. All you need is milk, eggs, and honey, and then you can add some flour or corn starch and some vanilla. Okay, all right, let’s just use up all …