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Hi, I’m Rob. Welcome to Math Antics.

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In this video, we’re going to learn about Polynomials.

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That’s a big math word for a really big concept in Algebra, so pay attention.

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Now before we can understand what polynomials are, we need to learn about what mathematicians call “terms”.

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In Algebra, terms are mathematical expressions that are made up of two different parts:

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a number part and a variable part.

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In a term, the number part and the variable part are multiplied together,

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but since multiplication is implied in Algebra,

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the two parts of a term are usually written right next to each other with no times symbol between them.

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The number part is pretty simple… it’s just a number, like 2 or 5 or 1.4

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And the number part has an official name… it’s called the “coefficient”.

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Now there’s another cool math word that you can use to impress your friends at parties!

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[party music, crowd noise]

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…and then I said, “That’s not my wife… that’s my coefficient!”

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[silence / crickets chirping]

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The variable part of a term is a little more complicated.

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It can be made up of one or more variables that are raised to a power.

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Like… the variable part could be 'x squared'. That’s a variable raised to a power.

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Or, the variable part could be just ‘y’.

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If you remember what we learned in our last video, you’ll realize that that also qualifies as a variable raised to a power.

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‘y’ is the same as ‘y’ to the 1st power.

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But since the exponent ‘1’ doesn’t change anything, we don’t need to actually show it.

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Or… the variable part of a term could be some tricky combination of variables that are raised to powers,

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like ‘x squared’ times ‘y squared’.

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…or ‘a’ times ‘b squared’ times ‘c cubed’.

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Terms can have any number of variables like that, but the good news is that most of the time,

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you’ll only need to deal with terms that have one variable. …or maybe two in complicated problems.

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Oh, and there’s one thing I should point out before we move on…

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if you have a term like 6y, even though it would be fine to do the multiplication the other way around and write y6,

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it’s conventional to always write the number part of the term first and the variable part of the term second.

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Okay, so that’s the basic idea of a term.

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But there’s a little more to terms that we’ll learn in a minute.

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First, let’s see how this basic idea of a term helps us understand the basic idea of a polynomial.

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A polynomial is a combination of many terms.

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It’s kind of like a chain of terms that are all linked together using addition or subtraction.

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The terms themselves contain multiplication, but each term in a polynomial must be joined by either addition or subtraction.

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And polynomials can be made from any number of terms joined together,

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but there are a few specific names that are used to describe polynomials with a certain number of terms.

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If there’s only one term (which isn’t really a chain) then we call it a “monomial” because the prefix “mono” means “one”.

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If there are just two terms, then we call it a “binomial” because the prefix “bi” means “two”,

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and if there are three terms, then we call it a “trinomial” since the prefix “tri” means “three”.

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Beyond three terms, we usually just say “polynomial” since “poly” means “many”,

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and in fact, it’s common to simply use the term “polynomial” even when there are just 2 or 3 terms.

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Okay, so that’s the basic idea of a polynomial.

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It’s a series of terms that are joined together by addition or subtraction.

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Now, let’s see a typical example of a polynomial that will help us learn a little more about terms: 3 ‘x squared’ plus ‘x’ minus 5

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How many terms does this polynomial have?

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Well, based on what we’ve learned so far, you’re probably not quit sure.

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If the terms are the parts that are joined together by addition or subtraction, then this should have three terms,

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but it looks like there’s something missing with the last two terms.

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This middle term is missing its number part, and this last term is missing its variable part.

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That doesn’t seem to fit with our original definition of a term. What’s up with that?

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Well, the middle term is easy to explain.

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There really is a number part there, but it’s just ‘1’.

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Do you remember how ‘1’ is always a factor of any number?

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But, since multiplying by ‘1’ has no effect on a number or variable, we don’t need to show it.

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So, if you see a term in a polynomial that has only a variable part, you know that the number part (or coefficient) of that term is just ‘1’.

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Okay, but what about this last term that’s missing its variable part?

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Well, that’s a little trickier. Do you remember in our last video about exponents in Algebra,

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we learned that any number or variable that’s raised to the 0th power just equals ‘1’?

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That means we can think of this last term as having a variable ‘x’ that’s being raised to the 0th power.

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Since that would always just equal ‘1’, it’s not really a variable in the true sense of the word,

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and it has no effect on the value of the term.

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But it makes sense, especially if you remember the other rule from the last video.

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That rule says that any number raised to the 1st power is just itself,

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which helps us see that this middle term is basically the same as ‘1x’ raised to the 1st power.

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Now do you see the pattern?

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Each term has a number part and each term has a variable part that is raised to a power: 0, 1 and 2.

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But since ‘x’ to the ‘0’ is just ‘1’,

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and ‘x’ to the ‘1’ is just ‘x’,

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and anything multiplied by ‘1’ is just itself,

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the polynomial gets simplified so that it no longer looks exactly like the pattern it comes from.

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Oh, and this last term… the one that doesn’t have a truly variable part…

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it’s called a CONSTANT term because its value always stays the same.

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Alright… Now that you know what a Polynomial is, let’s talk about an important property of terms and polynomials called their “degree”.

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Now that might sound like the units we use to measure temperature or angles, but the degree we’re talking about here is different.

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The degree of a term is determined by the power of the variable part.

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For example, in this term, since the power of the variable is 4, we say that the degree of the term is 4, or that it’s a 4th degree term.

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And in this term, the power of the variable is 3, so it’s a 3rd degree term.

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Likewise, this would be a 2nd degree term and this would be a 1st degree term.

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Oh, and I suppose you could call a term with no variable part a “zero degree” term,

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but it’s usually just referred to as a “constant term”.

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Things are a little more complicated when you have terms with more that one variable.

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In that case, you add up the powers of each variable to get the degree of the term.

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Since the powers in this term are 3 and 2, it’s a 5th degree term because 3 + 2 = 5.

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Okay, but why do we care about the degree of terms?

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Well, it’s because polynomials are often referred to by the degree of their highest term.

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If a polynomial contains a 4th degree term (but no higher terms), then it’s called a “4th degree” polynomial.

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But if its highest term is only a 2nd degree term, then it’s called a “2nd degree” polynomial.

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Another reason that we care about the degree of the terms is that it helps us decide the arrangement of a polynomial.

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We arrange the terms in a polynomial in order from the highest degree to the lowest.

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…ya know, cuz, mathematicians like to keep things organized…

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[mumbeling] …nice… let’s see…double check…

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For example, this polynomial (which has 5 terms)

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should be rearranged so that the highest degree term is on the left, and the lowest degree term is on the right.

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But of course, not every polynomial has a term of every degree.

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This is a 5th degree polynomial, but it only has 3 terms.

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We should still put them in order from highest to lowest, even though it has terms that are missing.

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So, the “4x to the fifth” should come first.

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And then the “minus 10x”.

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And finally, the “plus 8”.

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By the way, it’s totally fine for a polynomial to have “missing” terms like that.

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And it’s sometimes helpful to think of those missing terms as just having coefficients that are all zeros.

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If the coefficient of a term is zero, then the whole term has a value of zero so it wouldn’t effect the polynomial at all.

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And speaking of coefficients…

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What if we need to re-arrange this polynomial so that its terms are in order from highest degree to lowest degree?

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The highest degree term is ‘5x squared’ but before we just move it to the front of the polynomial,

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it’s important to notice that it’s got a minus sign in front of it.

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Normally when we see a minus sign, we think of subtraction, but when it comes to polynomials,

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it’s best to think of a minus sign as a NEGATIVE SIGN that means the term right after it has a negative value (or a negative coefficient).

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In fact, instead of thinking of a polynomial as having terms that are added OR subtracted,

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it’s best to think of ALL of the terms as being ADDED,

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but that each term has either a POSITIVE or a NEGATIVE coefficient which is determined by the operator right in front of that term.

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For example, if you have this Polynomial, you should treat it as if all of the terms are being added together,

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and use the sign that’s directly in front of each term to tell you if it’s a positive or a negative term.

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This first term has a coefficient of ‘negative 4’, so it’s a negative term.

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The next term has a coefficient of ‘positive 6’, so it’s positive.

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The next term has a coefficient of ‘negative 8’, so it’s negative.

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And the constant term is just ‘positive 2’.

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And recognizing positive and negative coefficients helps us a lot when

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rearranging polynomials that have a mixture of positive and negative terms like our example here.

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If you think of the negative sign in front of the ‘5x squared’ term as part of its coefficient,

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then you’ll realize that when we move it to the front of the polynomial, the negative sign has to come with it.

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It has to come with it because it’s really a NEGATIVE term.

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If we don’t bring the negative sign along with it, we’ll be changing it into a positive term

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which would actually change the value of the polynomial.

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And in addition to helping us re-arrange them,

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treating a polynomial as a combination of positive and negative terms will be very helpful when we need to simplify them,

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which just so happens to be the subject of our next basic Algebra video.

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Alright, we’ve learned a LOT about polynomials in this video,

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and if you’re a little overwhelmed, don’t worry… it might just take some time for it all to make sense.

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Remember, you can always re-watch this video a few times,

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and doing some of the practice problems will help it all sink in.

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As always, thanks for watching Math Antics, and I’ll see ya next time.