WEBVTT mathematics/algebra-1/smith
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Welcome back to www.educator.com.
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In this lesson we are going to take a look at how you can add and subtract polynomials.
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Before we get too far though or have to say what polynomials are and how we can classify them.
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I think we will finally get into adding and subtracting them.
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To watch for along the way, I will cover how you can evaluate polynomials for several different values.
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Recall some vocabulary that we had earlier.
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A term is a piece that is either connected using addition or subtraction.
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In this little example down below, this expression I have four terms.
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A coefficient is the number in front of the variable.
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The coefficient of this first term would be a 4 and I have a coefficient of 6, - 5 and 8 would be a coefficient.
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We often like to organize things from the highest power to the smallest power.
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The highest power here, it is a special name we will call this the leading coefficient.
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This will be important terms you will often here me use later on when talking about polynomials.
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Recall that things are like terms if they have the exact combination of variables with the same exponents.
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In this giant list here, I have lots and lots of examples of like terms.
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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.
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I have 19, 14 but it does not matter that part.
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What does matter is that I they both have an m³.
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The next two have both y⁹ and if you have more than one variable in there then both of those variables better matchup.
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Both of them have a single x and they both have a y².
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If we understand polynomials and understand terms then you can start understanding polynomials.
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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.
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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.
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I will pick these over and make sure that they fit the definition.
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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.
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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.
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In fact, there are no fractions there and we do not have any variables in the denominator.
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That is why that one is a polynomial.
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I will put in this next example to highlight that you could have coefficient that are fractions
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but the important part is that we do not have variables in the denominator, that would make it not a polynomial.
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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.
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Even the next one is a good example of this.
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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.
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Now a lot of things get to be a polynomial, even very small things.
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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.
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All of the exponents are nice whole powers and there are no variables in the denominator.
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You can have even very, very short polynomials.
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This one is just the number 5 has no variables, or even consider it x⁰.
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Compare these ones to things that are not polynomials and watch how they break the definition in some sort of way.
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In this first one is not very big, it is 1 ÷ x and it is not polynomial because we are dividing by x.
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We do not want that x in the bottom.
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This next one is not a polynomial because of its exponents.
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It has an exponent of ½ and another exponent of 1/3 and because of those exponents, it is not a polynomial.
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The next one is a little tricky.
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It looks like it should be a polynomial, I mean I have 1, 2x to the first and 3x².
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The reason why this is not a polynomial has to do with this little…
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That indicates that this keeps going on and on forever.
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In order to be a polynomial, it should stop.
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It should be finite somewhere.
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That one is not a polynomial.
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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.
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That should give you a better idea of when something is a polynomial or not a polynomial.
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We can start to classify the types of polynomials we have by looking at two aspects.
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One is how many terms they have.
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If it only has one term we would call that a monomial.
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In example 3x, it only has a single term, it is a monomial.
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If it has two terms then we can call that one a binomial, think of like a bicycle or something like that, two terms.
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If I have 3 terms we will go ahead and call it a trinomial.
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Usually if it has four or more terms you will get a little bit lazy we just usually call those as polynomials.
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Technically, all of these are examples of polynomials but they are just a very specific type of polynomial.
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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.
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In addition to talking about the monomial and binomial, that fun stuff, you can also talk about the degree of a polynomial.
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What the degree is, it is the highest power of any nonzero terms.
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You are looking for that biggest exponent.
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In this first one here, you can see that the largest exponent is 2, we would say that this is a 2nd degree polynomial.
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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.
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Now you can combine these two schemes together and get specific on the types of polynomials you are talking about.
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Not only for that first one can I say it is a 2nd degree because it has the largest power of 2 in there,
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but I can say it is a 2nd degree binomial because it has two terms in it.
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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.
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That is because we have 1, 2, 3 terms present in a polynomial.
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To evaluate a polynomial it is a lot like evaluating functions.
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We simply take the value that we are given and we substitute it for all copies of the variables.
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Let us give that a try with 2y³ + 8y - 6 and we want to evaluate it for y = -1.
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I will go through and everywhere I see a copy of y, we will put in that -1.
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Let us go ahead and work on simplifying this.
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-1³ would be -1 × -1× -1,that would simply be -1.
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8 × -1 would be -8 and then we have -6.
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Continuing on, I have -2 – 8 – 6 = -16.
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If you are evaluating it, just take that value and substitute it in for all copies of that variable.
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Onto what we wanted to, adding and subtracting polynomials.
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When we get into addition and subtraction, what we are looking to do is add or subtract the like terms present in the polynomial.
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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.
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Here I want to add the following two polynomials.
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The way I’m going to do this is I'm going to simply highlight which terms are like terms.
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Here is 4x³ and 6x³ those are like terms - 3x² 2x², those are like.
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2x and - 3x and so one by one we will take these like terms and simply put them together.
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4x³ and 6x³ = 10x³.
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-3x² and 2x² would be -1x².
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2x - 3x =- 1x.
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You can see I have all of the parts there and it looks like I am left with a 3rd degree trinomial.
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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.
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If you do it this way, you want to make sure you line up what are your like terms.
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Put your x³ on your x³, your x² on your x² and you x on your x.
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It is essentially the same idea and that you go through adding all of the terms as you go along.
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I will start over on the right side of this one, 2x and I'm adding - 3x, there is - x for that one.
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Onto the next set of terms, - 3x² + 2x² = - x² and 4x³ and 6x³ = 10x³.
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You can see you get exactly the same answer but just use whatever method you are more familiar with.
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Let us try some examples.
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In this one we want to determine if these are polynomials or not.
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Let us try that first one.
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I see I have 2x + 3x² – 8x³.
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The things we are watching out for is, one does it stop.
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I do not see any… out here, I know this is finite, that looks pretty good.
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I do not see any variables in the denominator, so that is good.
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There are no fractions with the x's in the bottom.
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All of the exponents here, the 1, 2, and the 3 all of those are nice whole numbers.
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This one is looking good.
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I will say that this one is a polynomial.
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2/x + 5/4x² – x³/6.
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In this one I can think I can see a problem right away.
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Notice how we have variables in the bottom and because of that I will say that this one is not a polynomial.
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Be careful and watch out for that criteria.
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This next example, we want to just go through and classify what types of polynomials these are.
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We use two criteria for this, we look at its degree and we will see how many terms it has.
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The first one, the largest power I can see in here is this 3.
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I will say this is a 3rd degree.
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It is a polynomial but let us be a little more specific.
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It has 1, 2, 3 terms, so I will say this is a 3rd degree trinomial.
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Let us try another one, the largest power here is 4, 4th degree.
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It only has 1, 2 terms, so it will be our binomial.
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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.
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This is a 3rd degree and now we count up all of its terms, 1, 2, 3, 4.
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Since it has four terms, I will just keep calling this one a polynomial.
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Let us get into adding the following polynomials together.
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The way I’m going to do this is I’m going to line them up, one on top of the other.
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Starting with the first one, I have 3x³, I do not have any x², I have 4x and I have 1.
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All of that would represent my first polynomial there.
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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.
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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.
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You can now have things all nice and lined up.
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Let us go ahead and add them one at a time.
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1 + 6 = 7, -4 + 6x =2x, 0x² + -3x² = -3x².
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One more, this has nothing to add to it so just 3x².
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This would be our completed polynomial after adding the two together.
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One last example is we are going to work on subtracting polynomials.
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With these ones, what I suggest is being very careful with your signs.
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You will see that I’m going to start with lining one on top of the other one.
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But I’m going to end up distributing my negative signs, I will turn this into an addition problem.
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Let us give it a try.
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The polynomial on the left is 7y² -11y + 8 and right below that is -3y² + 4y + 6.
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Now comes the important part, we are subtracting these polynomials so I will put a giant minus sign up front.
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Before getting too far, I could go through and try and subtract this term by term,
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but it is much easier to distribute my negative sign and just look at this like an addition problem.
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My top polynomial will stay unchanged and we will leave that as it is.
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After distributing the bottom, here is what we get negative × - 3y = 3y², negative × 4y = -4y and for the very last one -6.
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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.
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8 + -6 = 2, -11y + -4y = -15y and then I have a 7y² + 3y² = 10y².
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This final polynomial represents the two being subtracted.
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Now you know a little bit more about polynomials and have put them together.
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Remember that you are just combining your like terms.
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