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# Polynomials intro

CCSS.Math:

## Video transcript

let's explore the notion of a polynomial so this seems like a very complicated word but if you break it down they'll start to make sense especially when we start to see examples of polynomials so the first part of this word let me underline it with poly this comes from Greek for many and you see poly a lot in the English language referring to the notion of many of something so in this case it's many no meals and no meal comes from Latin from the Latin Roman for name so you could view this as many names but in the mathematical context it's really referring to many terms and we're going to talk and a little bit about what a term really is but I get a tangible sense of what are polynomials and what are not polynomials let me give you some examples and then we could write some may be more formal rules for them so an example of a polynomial could be 10 X to the seventh power minus 9 x squared plus 15 X to the third plus 9 this is a polynomial another example of a polynomial 9a squared minus 5 even if I just have one number even if I were to just write the number six that can officially be considered a polynomial if I were to write 7x squared minus three let me do is another variable 7y squared minus 3y plus pi that 2 would be a polynomial so these are examples of polynomials what are examples of things that are not polynomials well if I were to replace the seventh power right over here with the negative seventh power so if I were to write 10x to the negative seven power minus nine x squared plus 15 X to the third power plus nine this would not be a polynomial so I think you might be sensing a rule here for what makes something upon no meal you have to have non-negative powers of your variable in each of the terms and I just use that word term so let me explain it because it will help me explain what a polynomial is a polynomial is something that is it that is made up of a sum of terms and so for example in this first polynomial the first term is 10 X to the seventh the second term is negative 9x squared the next term is 15 X to the third and then the last term maybe you could say the fourth term is nine and you can see something let me underline these so these are all these are all terms this is a four term polynomial right over here and you could say hey wait this thing you wrote in red this also has four terms but we have to put a few more rules for to officially be a polynomial especially a polynomial in one variable each of those terms are going to be made up of a coefficient this is the thing that multiplies the variable to some power so in the first term the coefficient is 10 and let me write this word down coefficient so another fancy word but it's just a thing that's multiplied in this case times the variable which is X to the seventh power so the Co the first coefficient is 10 the next coefficient and actually let me be careful here because the second coefficient here is negative 9 so we are looking at coefficients the third coefficient here is 15 and you can view this fourth term or this fourth number as the coefficient because this could be re-written as instead of just writing as 9 you could write it as 9x to the 0 power and then it looks a little bit clearer like a coefficient so in general a polynomial is the sum of a finite number of terms where each term has a coefficient which I could represent with the letter a being multiplied by a variable being raised to a non-negative integer power so this right over here is the coefficient it can be if we're dealing well I don't want to get too technical could we give a positive negative number it could be any real number we have our variable and then the exponent here has to be non-negative non-negative integer so here the reason why what I wrote in red is not a polynomial is because here I have an exponent that is a negative integer let's give some other examples of things that are not polynomials so if I were to change the second one to sort of 9a squared if I wrote it as 9a to the one half power minus five this is not a polynomial because this exponent right over here it is no longer an integer it's one half and this is the same thing as nine times the square root of a minus five this also would not be a polynomial or if I were to write nine a to the a power minus five also not a polynomial because here the exponent is a variable it's not a non-negative integer so all of these are examples of polynomials so there's a few more pieces of terminology that are valuable to know polynomials is the general polynomial is a general term for one of these expressions that has multiple terms a finite number so not an infinite number and each of the terms has this form but there's more specific terms for when you have only one term or two terms or three terms so when you have one term it's called a monomial so this is a monomial this is an example of a monomial which we could write at 6 x2 0 but you could also another example of a monomial might be 10 10 Z to the 15th power that's also a monomial your coefficient could be pi pi whoops it could be pi so we could write pi x times B to the fifth power any of these would be monomials so it's a binomial or binomials where you have two terms monomial mono for one one term binomial is you have two terms so this right over here is a binomial by no meal you have two terms and all of these are polynomials but these are sub classifications so it's by no Milly of one two terms another example of a binomial would be would be 3y to the third plus 5y once again you have two terms that have this form right over here now you'll also hear the term trinomial well trinomials when you have three terms try no meal and this right over here is an example this is the first term this is the second term and this is the third term now the next word that you will hear often in the context with polynomials is the notion of the degree of a polynomial and you might hear people say what is the degree of a polynomial or what is the degree of a given term of a polynomial so let's start with the degree of a given term so let's go to this polynomial over here we have this first term 10 X to the seventh the degree is the power that we're raising the variable to so this is a 7th degree term the second term is a second degree term the third term is a third degree term and you could view this constant term which is really just nine you could do that as sometimes people say the constant term sometimes people will say the zeroth degree term now people are talking about the degree of the entire polynomial they're going to say well what is the degree of the highest term or the term with the highest degree that degree will be the degree of the entire polynomial so this first polynomial this is a seventh degree polynomial this one right over here is a second degree polynomial because it has a second degree term and that's the highest degree term this right over here is a third degree you could even say third degree binomial because it's highest degree term has degree 3 if they said 5 Y to the seventh instead of 5 Y well then it would be a seventh degree binomial this right over here is a 15 degree monomial this is a second-degree trinomial now the last thing I a few more things I will introduce you to is the idea of a leading term and a leading coefficient let me write this down the notion of what it means to be leading well it usually means it can mean whatever is the first term or the coefficient if you're saying leading term it's the first term and if you're saying leading coefficient is the coefficient in the first term but it's oftentimes associated with a polynomial being written in standard form so standard form standard form is where you write the terms in degree order starting with the highest degree term so for example what I have up here this is not in standard form because I do have the highest degree term first but then I should go to the next highs which is the X to the third but here I wrote x squared next so this is not standard if I wanted to write it in standard form it would be 10 X to the seventh power which is the highest degree term it's a degree 7 then 15 X to the third so + 15 X to the third which is the next highest degree then negative 9x squared is the next highest degree term and then the lowest degree term here is plus 9 or plus 9x to 0 now this is in standard form I've written the terms in order of decreasing degree with the highest degree first and here it's clear that your leading term is 10 X to the seventh because it's the first one and our leading coefficient here is the number 10 so there's a lot in that video but hopefully that the notion of a polynomial isn't seeming too intimating at this point and these are really useful words to be familiar with as you continue on on your math journey