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

Factoring using polynomial division: missing term | Algebra 2 | Khan Academy


3m read
·Nov 11, 2024

We're told the polynomial ( p(x) ) which is equal to this has a known factor of ( x + 6 ). Rewrite ( p(x) ) as a product of linear factors. Pause this video and see if you can have a go at that.

All right, now let's work on this together. Because they give us one of the factors, what we can do is say, "Hey, what happens if I divide ( x + 6 ) into ( p(x) )? What do I have left over?" It looks like I'm still going to have a quadratic, and then I'll probably have to factor that somehow to get a product of linear factors. So let's get going.

If I were to try to figure out what ( x + 6 ) divided into ( x^3 + 9x^2 ), and now we're going to have to be careful. You might be tempted to just write -108 there, but then this gets tricky because you have your third-degree column, your second-degree column, you need your first-degree column, but you just put your zero-degree, your constant column here.

So to make sure we have good hygiene, we could write ( + 0x ), and I encourage you to actually always do this if you're writing out a polynomial so that you don't skip that place, so to speak, -108.

And so then you say, "All right, let's look at the highest degree terms." ( x ) goes into ( x^3 ) ( x^2 ) times. ( x^2 ) times ( 6 ) is ( 6x^2 ). ( x^2 ) times ( x ) is ( x^3 ). We want to subtract. We've done this multiple times, so I'm going a little bit faster than normal. Those cancel out.

( 9x^2 - 6x^2 = 3x^2 ). Bring down that ( 0x ). And then how many times does ( x ) go into ( 3x^2 )? Well, it goes ( 3x ) times, and we would write it in this column. Notice if we didn't keep this column for our first-degree terms, we'd be kind of confused where to write that ( 3x ) right about now.

And so ( 3x ) times ( 6 ), I should say, is ( 18x ). ( 3x ) times ( x ) is ( 3x^2 ). We want to subtract what we have in that, I guess that color is move light purple, not sure. And so we get ( 3x^2 )'s cancel out, and then ( 0x - 18x = -18x ). Bring down that ( -108 ).

And so then we have ( x ) goes into ( -18x ) ( -18 ) times. ( -18 ) times ( 6 ) is ( -108 ). That's working out nicely. ( -18 ) times ( x ) is ( -18x ), and then we want to subtract what we have in this not so pleasant brown color.

And so I will multiply them both by negative, and so I am left with zero; everything just cancels out. And so I can rewrite ( p(x) ). I can rewrite ( p(x) ) as being equal to ( x + 6 \times (x^2 + 3x - 18) ).

But I'm not done yet because this is not a linear factor; this is still quadratic. So let's see, can I think of two numbers that add up to ( 3 ) and then when I multiply I get ( -18 )? So they'll need different signs, and then the obvious one is positive ( 6 ) and negative ( 3 ).

And if that what I just did seems like voodoo to you, I encourage you to review factoring polynomials. But this I can rewrite because negative ( 6 + ) or actually I should say positive ( 6 + (-3) ) is equal to ( 3 ), and then positive ( 6 \times negative ( 3 ) is equal to ( -18 ).

So I can rewrite this as ( x + 6 \times (x + 6) \times (x - 3) ). And so there we have it; we have a product of linear factors, and we are done.

More Articles

View All
Asking Billionaires How They Got Rich! (Houston)
Who am I here with today? Damon John. Kendra Scott, are you a business owner? I am. I’m one of only 20 female founders in the United States that have founded a billion-dollar brand. So you founded a billion-dollar company? A billion-dollar company, with a…
Where Does the Waste Go?: A Day in the Life of a Scientist | Continent 7: Antarctica
[Music] Definitely the worst part about Antarctica. So we don’t leave anything behind here in the environment. The New Zealand program actually is very thorough in doing that and it’s not that bad as it sounds. So I disagree. Uh, yeah, some disagree. Actu…
The Market Revolution - part 2
So we’ve been talking about the market revolution in the United States, which was this period in the first half of the 19th century where the way that Americans did business really changed. It changed in a number of ways. The kinds of work that people did…
Why Do You Make People Look Stupid?
Hey Youtube, you said you wanted to talk. What’s up? Why do you go around making other people look stupid? What do you mean? What’s water made of? Water. Hahaha, what makes water? Water. Ok, what elements does it take to make water? H2O. So what …
The mindset that's changing my life
I feel like everybody at some point in their life has met somebody who was truly inspiring. You know, they seem to have their life figured out. They are determined; they can carve out their own destiny. They create their own luck. On the flip side, a lot…
Spider vs Penis (Priapism) - Smarter Every Day 98
Alright, so this video may not be appropriate for kids, and it is, uh… It’s disturbing on several different levels. Especially if you’re a man… So, you know, on Smarter Every Day, I try to keep everything very intelligent and respectful, but this video is…