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

Dividing polynomials by x (no remainders) | Algebra 2 | Khan Academy


3m read
·Nov 11, 2024

Let's say someone walks up to you on the street and they give you this expression:

x squared plus 7x plus 10 divided by x plus 2.

And they say, “See if you could simplify this thing.”

So, pause this video and see if you can do that.

One way to think about it is: what is x squared plus 7x plus 10 divided by x plus 2?

What is that going to be?

All right, now there are two ways that you could approach this. One way is to try to factor the numerator and see if it has a factor that is common to the denominator. So, let's try to do that.

We've done this many, many times. If this looks new to you, I encourage you to review factoring polynomials other places on Khan Academy.

But what two numbers add up to seven and when you multiply them you get ten?

Well, that would be two and five.

So we could rewrite that numerator as (x + 2)(x + 5).

And then, of course, the denominator you still have (x + 2).

Then we clearly see we have a common factor.

As long as x does not equal negative 2, because if x equals negative 2, this whole expression is undefined.

This is because then you get a 0 in the denominator.

So as long as x does not equal negative 2, then we can divide the numerator and the denominator by (x + 2).

Once again, the reason why I put that constraint is we can't divide the numerator and denominator by zero.

So for any other values of x, this (x + 2) will be non-zero, and we could divide the numerator and the denominator by that and they would cancel out.

We would just be left with x + 5.

So another way to think about it is that our original expression could be viewed as x + 5 for any x that is not equal to negative 2.

Now, the other way that we could approach this is through algebraic long division, which is very analogous to the type of long division that you might remember from, I believe, it was fourth grade.

So what you do is you say, “All right, I'm going to divide (x + 2) into (x squared + 7x + 10)."

In this technique, you look at the highest degree terms.

So you have an x there and an x squared there, and you say, “How many times does x go into x squared?”

Well, it goes x times.

Now you would write that in this column because x is just x to the first power.

You could view this as the first degree column.

It's analogous to the place values that we talk about when we first learn numbers or how we regroup or talk about place value.

But here you can view it as degree places or something like that.

Then you take that x and you multiply it times this entire expression.

So x times 2 is 2x.

Put that in the first degree column.

x times x is x squared.

What we want to do is subtract these things in yellow from what we originally had in blue.

So we could do it this way, and then we will be left with:

7x minus 2x is 5x, and then x squared minus x squared is just zero.

Then we can bring down this plus 10.

Once again we look at the highest degree term.

x goes into 5x five times.

That's a zero degree; it's a constant.

So I'll write it in the constant column.

5 times 2 is 10, and 5 times x is 5.

Then I'll subtract these from what we have up here, and notice we have no remainder.

What's interesting about algebraic long division, we'll probably see in another video or two, is that you can actually have a remainder.

Those are going to be situations where just the factoring technique alone would not have worked.

In this situation, this model would have been easier.

But this is another way to think about it:

You say, “Hey, look, (x + 2)(x + 5) is going to be equal to this.”

Now, if you wanted to rewrite this expression the way we did here and say, “Hey, this expression is equal to x + 5,” we would have to constrain the domain.

You'd say, “Hey, for all x's not equaling negative 2 for these to be completely identical expressions.”

More Articles

View All
Knowing Yourself
I think that one of the most important fundamental ingredients to being happy in life and being successful is to be realistic about yourself, your preferences, and also your strengths and weaknesses that everybody has. I think the system, particularly th…
Homeroom with Sal & Dan Tieu & Sophie Turnbull - Wednesday, August 19
Hi everyone, Sal here from Khan Academy. Welcome, uh, again for joining our homeroom live stream. Uh, before we get into what’s going to be a really fun conversation with some internal Khan Academy team members to talk about all the new things we have for…
2011 Calculus AB Free Response #1 parts b c d | AP Calculus AB | Khan Academy
Alright, now let’s tackle Part B. Find the average velocity of the particle for the time period from zero is less than or equal to T is less than or equal to 6. So our average velocity, that’s just going to be our change in position, which we could view …
Khan Academy Talent Search 2016
Hi, this is Sal Khan, founder of the KH Academy, and I just wanted to announce our second annual Talent Search. As you may know, KH Academy, we’re a non-profit with the mission of a free, world-class education for anyone, anywhere. Part of that is creatin…
Worked example: distance and displacement from position-time graphs | AP Physics 1 | Khan Academy
In other videos, we’ve already talked about the difference between distance and displacement, and we also saw what it meant to plot position versus time. What we’re going to do in this video is use all of those skills. We’re going to look at position vers…
Determining if a function is invertible | Mathematics III | High School Math | Khan Academy
[Voiceover] “F is a finite function whose domain is the letters a to e. The following table lists the output for each input in f’s domain.” So if x is equal to a, then if we input a into our function, then we output -6. f of a is -6. We input b, we get …