Algebra 1 Adding & Subtracting Polynomials

Welcome back to www.educator.com.0000

In this lesson we are going to take a look at how you can add and subtract polynomials.0002

Before we get too far though or have to say what polynomials are and how we can classify them.0009

I think we will finally get into adding and subtracting them.0015

To watch for along the way, I will cover how you can evaluate polynomials for several different values.0018

Recall some vocabulary that we had earlier.0029

A term is a piece that is either connected using addition or subtraction.0033

In this little example down below, this expression I have four terms.0040

A coefficient is the number in front of the variable.0051

The coefficient of this first term would be a 4 and I have a coefficient of 6, - 5 and 8 would be a coefficient.0055

We often like to organize things from the highest power to the smallest power.0068

The highest power here, it is a special name we will call this the leading coefficient.0073

This will be important terms you will often here me use later on when talking about polynomials.0083

Recall that things are like terms if they have the exact combination of variables with the same exponents.0092

In this giant list here, I have lots and lots of examples of like terms.0100

This first one is an example of like terms because they both have an m³ and notice it has no difference what the coefficient out front is.0106

I have 19, 14 but it does not matter that part.0117

What does matter is that I they both have an m³.0121

The next two have both y⁹ and if you have more than one variable in there then both of those variables better matchup.0127

Both of them have a single x and they both have a y².0136

If we understand polynomials and understand terms then you can start understanding polynomials.0144

A polynomial is a term or a finite sum of terms in which all the variables have whole number exponents and no variables up here in the denominator.0150

To make this a little bit more clear, I have many different examples of polynomials and many different examples of things that are not polynomials.0160

I will pick these over and make sure that they fit the definition.0168

This first one here, I know that it is a polynomial as I can see that has a finite number of terms that means that stops eventually.0174

If I look at all the exponents present like the two, then all of those are nice whole numbers and I do not see anything in the denominator.0182

In fact, there are no fractions there and we do not have any variables in the denominator.0191

That is why that one is a polynomial.0196

I will put in this next example to highlight that you could have coefficient that are fractions0201

but the important part is that we do not have variables in the denominator, that would make it not a polynomial.0206

It is okay if we have more than one variable like start mixing around m and p, just as long as the exponents on those state nice whole numbers.0216

Even the next one is a good example of this.0227

I have y and x, but I have a finite number and eventually stops and it looks like I do not have any variables in the bottom.0229

Now a lot of things get to be a polynomial, even very small things.0239

For example, something like 4x is a type of polynomial, it is not a very big polynomial and has exactly one term, but is finite.0245

All of the exponents are nice whole powers and there are no variables in the denominator.0253

You can have even very, very short polynomials.0259

This one is just the number 5 has no variables, or even consider it x⁰.0261

Compare these ones to things that are not polynomials and watch how they break the definition in some sort of way.0267

In this first one is not very big, it is 1 ÷ x and it is not polynomial because we are dividing by x.0276

We do not want that x in the bottom.0282

This next one is not a polynomial because of its exponents.0286

It has an exponent of ½ and another exponent of 1/3 and because of those exponents, it is not a polynomial.0290

The next one is a little tricky.0299

It looks like it should be a polynomial, I mean I have 1, 2x to the first and 3x².0301

The reason why this is not a polynomial has to do with this little…0307

That indicates that this keeps going on and on forever.0311

In order to be a polynomial, it should stop.0315

It should be finite somewhere.0318

That one is not a polynomial.0319

Other things that we want to watch out for is we do not have any of our variables and roots and none of those exponents should be negative.0323

That should give you a better idea of when something is a polynomial or not a polynomial.0334

We can start to classify the types of polynomials we have by looking at two aspects.0342

One is how many terms they have.0348

If it only has one term we would call that a monomial.0351

In example 3x, it only has a single term, it is a monomial.0357

If it has two terms then we can call that one a binomial, think of like a bicycle or something like that, two terms.0363

If I have 3 terms we will go ahead and call it a trinomial.0374

Usually if it has four or more terms you will get a little bit lazy we just usually call those as polynomials.0384

Technically, all of these are examples of polynomials but they are just a very specific type of polynomial.0391

Let me write that one with a bunch of different terms, 5x⁴ - 3x³ + x² - x +7 that would be a good example of a polynomial.0398

In addition to talking about the monomial and binomial, that fun stuff, you can also talk about the degree of a polynomial.0413

What the degree is, it is the highest power of any nonzero terms.0421

You are looking for that biggest exponent.0426

In this first one here, you can see that the largest exponent is 2, we would say that this is a 2nd degree polynomial.0429

In the next one, just off to the right, the largest power in there is a 3, so this would be a 3rd degree polynomial.0440

Now you can combine these two schemes together and get specific on the types of polynomials you are talking about.0452

Not only for that first one can I say it is a 2nd degree because it has the largest power of 2 in there,0459

but I can say it is a 2nd degree binomial because it has two terms in it.0465

With the other one in addition to saying it is a 3rd degree polynomial, I can take a little further and say it is a 3rd degree trinomial.0473

That is because we have 1, 2, 3 terms present in a polynomial.0484

To evaluate a polynomial it is a lot like evaluating functions.0494

We simply take the value that we are given and we substitute it for all copies of the variables.0498

Let us give that a try with 2y³ + 8y - 6 and we want to evaluate it for y = -1.0504

I will go through and everywhere I see a copy of y, we will put in that -1.0513

Let us go ahead and work on simplifying this.0527

-1³ would be -1 × -1× -1,that would simply be -1.0532

8 × -1 would be -8 and then we have -6.0539

Continuing on, I have -2 – 8 – 6 = -16.0546

If you are evaluating it, just take that value and substitute it in for all copies of that variable.0557

Onto what we wanted to, adding and subtracting polynomials.0568

When we get into addition and subtraction, what we are looking to do is add or subtract the like terms present in the polynomial.0573

There are two ways you should go about this and both of them are perfectly valid so you will use whichever method you are more comfortable with.0581

Here I want to add the following two polynomials.0590

The way I’m going to do this is I'm going to simply highlight which terms are like terms.0593

Here is 4x³ and 6x³ those are like terms - 3x² 2x², those are like.0599

2x and - 3x and so one by one we will take these like terms and simply put them together.0610

4x³ and 6x³ = 10x³.0619

-3x² and 2x² would be -1x².0626

2x - 3x =- 1x.0633

You can see I have all of the parts there and it looks like I am left with a 3rd degree trinomial.0640

The other way that you can do this, if you are a little bit more comfortable with it, is you can take one polynomial over the other polynomial.0645

If you do it this way, you want to make sure you line up what are your like terms.0660

Put your x³ on your x³, your x² on your x² and you x on your x.0665

It is essentially the same idea and that you go through adding all of the terms as you go along.0672

I will start over on the right side of this one, 2x and I'm adding - 3x, there is - x for that one.0678

Onto the next set of terms, - 3x² + 2x² = - x² and 4x³ and 6x³ = 10x³.0688

You can see you get exactly the same answer but just use whatever method you are more familiar with.0701

Let us try some examples.0711

In this one we want to determine if these are polynomials or not.0712

Let us try that first one.0718

I see I have 2x + 3x² – 8x³.0719

The things we are watching out for is, one does it stop.0723

I do not see any… out here, I know this is finite, that looks pretty good.0727

I do not see any variables in the denominator, so that is good.0732

There are no fractions with the x's in the bottom.0736

All of the exponents here, the 1, 2, and the 3 all of those are nice whole numbers.0740

This one is looking good.0747

I will say that this one is a polynomial.0749

2/x + 5/4x² – x³/6.0757

In this one I can think I can see a problem right away.0762

Notice how we have variables in the bottom and because of that I will say that this one is not a polynomial.0765

Be careful and watch out for that criteria.0776

This next example, we want to just go through and classify what types of polynomials these are.0782

We use two criteria for this, we look at its degree and we will see how many terms it has.0788

The first one, the largest power I can see in here is this 3.0794

I will say this is a 3rd degree.0798

It is a polynomial but let us be a little more specific.0811

It has 1, 2, 3 terms, so I will say this is a 3rd degree trinomial.0814

Let us try another one, the largest power here is 4, 4th degree.0826

It only has 1, 2 terms, so it will be our binomial.0838

One more to classify, this one has a bunch of different exponents, but of course the largest one is the only one we are interested in.0847

This is a 3rd degree and now we count up all of its terms, 1, 2, 3, 4.0856

Since it has four terms, I will just keep calling this one a polynomial.0870

Let us get into adding the following polynomials together.0883

The way I’m going to do this is I’m going to line them up, one on top of the other.0886

Starting with the first one, I have 3x³, I do not have any x², I have 4x and I have 1.0892

All of that would represent my first polynomial there.0905

Below that I want to write the second polynomial but I want to lineup the terms, -3x² I will put it under the other x².0910

I have a 6x I will put that under the other x and the 6 I will go ahead and put that with the 1.0922

You can now have things all nice and lined up.0931

Let us go ahead and add them one at a time.0934

1 + 6 = 7, -4 + 6x =2x, 0x² + -3x² = -3x².0938

One more, this has nothing to add to it so just 3x².0961

This would be our completed polynomial after adding the two together.0967

One last example is we are going to work on subtracting polynomials.0977

With these ones, what I suggest is being very careful with your signs.0981

You will see that I’m going to start with lining one on top of the other one.0985

But I’m going to end up distributing my negative signs, I will turn this into an addition problem.0989

Let us give it a try.0995

The polynomial on the left is 7y² -11y + 8 and right below that is -3y² + 4y + 6.0996

Now comes the important part, we are subtracting these polynomials so I will put a giant minus sign up front.1015

Before getting too far, I could go through and try and subtract this term by term,1023

but it is much easier to distribute my negative sign and just look at this like an addition problem.1028

My top polynomial will stay unchanged and we will leave that as it is.1036

After distributing the bottom, here is what we get negative × - 3y = 3y², negative × 4y = -4y and for the very last one -6.1043

We can take care of this as an addition problem and we know that the subtraction is taken care of because we put it into all of our terms.1060

8 + -6 = 2, -11y + -4y = -15y and then I have a 7y² + 3y² = 10y².1071

This final polynomial represents the two being subtracted.1098

Now you know a little bit more about polynomials and have put them together.1105

Remember that you are just combining your like terms.1109

Thank you for watching www.educator.com.1112