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
"The 4 THINGS Poor People DO That The RICH DON'T!" | Kevin O'Leary
If you’re a CEO and you’re just driven by business, which you know entrepreneurs really are, you’ve got to find a passion. She wanted to diversify her risk, is what she wanted. Because she didn’t, she knew you were great, but she didn’t know which one of …
Analyzing mistakes when finding extrema (example 1) | AP Calculus AB | Khan Academy
Pamela was asked to find where ( h(x) = x^3 - 6x^2 + 12x ) has a relative extremum. This is her solution. So, step one, it looks like she tried to take the derivative. Step two, she tries to find the solution to find where the derivative is equal to zero…
The Japanese Government Wants You to Date | Explorer
[music playing] FRANCESCA FIORENTINI (VOICEOVER): Here in the Japanese countryside, some of Japan’s most eligible bachelors are waiting to meet their mates. The mayor is here. Parents are here. Eligible bachelors and bachelorettes are here. FRANCESCA FI…
Advantages Of A First-Time Founder
First-time founders can actually take more risk on the ideas that they pick because they don’t have other startup friends, or they don’t care as much. They’re just working on stuff they find interesting. I love that they have nobody to impress, basically.…
Timeline of The Most Important Philosophical Ideas, I guess
We’re all pretty used to rain. We’re either prepared for it with an umbrella or raincoat, or just get wet. Rarely does it genuinely upset us. But what about when it rains for days and the streets flood so you can’t go outside? Or when you realize you can’…
Peter Lynch: How to invest in 2023
If you want to learn how to get rich investing in the stock market, Peter Lynch is someone you need to be learning from. Lynch has arguably the best track record of any stock picker that has managed large amounts of money during his time running the famou…