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Algebra I: High School20 chapters | 168 lessons | 1 flashcard set

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Lesson Transcript

Instructor:
*Yuanxin (Amy) Yang Alcocer*

Amy has a master's degree in secondary education and has taught math at a public charter high school.

In this video lesson, you will learn how to handle and work with polynomials that have more than one variable. This is important because in math you sometimes have to work with several variables at the same time.

In math, you won't always have the luxury of working with equations and functions that only have one variable. See, sometimes you will have equations and functions that have more than one variable.

In this video lesson, we will talk about what you can do when you have a polynomial with several variables. Recall that a **polynomial** is a function made up of terms connected by either a plus or a minus. Your terms are made up of a coefficient, the number part, and possibly a variable, the letter part.

You know very well how to evaluate the polynomial 4*x* + 5*x*. You know about like terms and so you can easily give an answer of 9*x*. And if we had 4*x* + 5*x* = 18, you can easily solve it for *x* to find that *x* = 2.

But, what will you do if we have a polynomial such as 4*x* + 2*y* + *z* = 0? This polynomial has several variables in it. Which one do you solve for? And, how do you do it? Keep watching and you will find out.

The good news is that in problems, they tell you which variable they want you to solve for. In our problem, we have three variables: *x*, *y*, and *z*. The problem will tell us which variable to solve for. Whenever you see several variables, don't always assume that you will solve for *x*. The problem might tell you to solve for *y*! Why don't we solve this problem for *y*? Let's see what happens.

To help me keep my variables straight, I like to think of them as different kinds of objects. I like food, so I think of my *x's* as French fries, my *y's* as burgers, and my *z's* as ice cream. This kind of visualization might also help you.

So for this problem, I am looking for burgers. I need to isolate my burgers so I can take a big bite out of them! I look at my polynomial and I see that I need to perform some subtraction to help me isolate my *y* variable.

We see that we have two terms that need to be moved to the other side. They have pluses in front of them, so to move them I need to subtract. If they have minuses in front of them, I would need to add to move them over. I am subtracting, so I will subtract first one term and then the other.

I remember that if I perform one operation on one side, I have to perform the same operation to the other. So, if I subtract 4*x* from one side, I also subtract it from the other. Doing this I get 2*y* + *z* = -4*x*.

Now, I need to subtract the *z* from both sides. Doing that I get 2*y* = -4*x* - *z*. Notice that the 4*x* and the *z* are not like terms so I can't combine them. Since I can't combine them, I leave them as they are and I just write them all out.

When you are working with several variables, don't worry about combining everything because when you have several variables you won't always be able to do that. It is perfectly okay to have a long answer with several variables.

I'm not done with my problem, though. I still need to get my *y* by itself. I see that a 2 is multiplying it.

To unhook it from the *y* I need to divide it. If instead of multiplication, I saw a division by 2, then I would multiply it by 2 to unhook the 2. But, I see multiplication, so dividing by 2 on both sides I get *y* = -2*x* - *z*/2. And I am done.

When working with several variables, you may very well end up with an answer that also contains variables. Just focus on the variable you are solving for and make sure that your answer doesn't have that variable in it.

Sometimes, the problem may give you numbers you can plug in. In this case, you can go ahead and solve for your variable and then plug in your numbers, or you can plug in your number first and then solve for your variable. Either way is fine.

So, say my problem told me that my *x* = 1 and my *z* = 0. The problem wants to know what my *y* equals when *x* = 1 and *z* = 0.

Well, since I've already solved my problem for *y*, I can plug in my *x* and *z* values into my answer to find out what *y* equals. So, wherever I see an *x* I will plug in a 1, and wherever I see a *z* I will plug in a 0. Let's see what we get.

*y* = -2(1) - 0/2

*y* = -2

Look at that! I now have an answer that is just a number.

So, what have we learned? We've learned that **polynomials** are functions made up of several terms. These terms are made up of coefficients, the number part, and variables, the letter part.

Sometimes you will need to solve or evaluate a polynomial with several variables in it. When this happens, you will need to first find out which variable the problem wants you to solve for. And then you solve for that variable.

If you see terms that don't contain your variable being added or subtracted from the term that has your variable, you perform the opposite operation to move it over to the other side. If you see addition, subtract. If you see subtraction, add. Now, if you see your variable being multiplied by a number, divide by it. If you see your variable being divided by a number, multiply by it.

And remember, whatever you do to one side, you also have to do to the other. If the problem gives you numbers to plug in for your letters, you can plug them in at the end or at the beginning.

Once you have completed this lesson, you should be able to solve a polynomial by isolating a variable.

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Algebra I: High School20 chapters | 168 lessons | 1 flashcard set

- How to Graph Cubics, Quartics, Quintics and Beyond 11:14
- How to Add, Subtract and Multiply Polynomials 6:53
- How to Divide Polynomials with Long Division 8:05
- How to Use Synthetic Division to Divide Polynomials 6:51
- Dividing Polynomials with Long and Synthetic Division: Practice Problems 10:11
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