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What is a polynomial function

welcome what I want to do is show

you how to determine or not really

determine but explain what a polynomial

function is so let's go through the

boring part first and then I'll I kind

of explain what is not a polynomial what

is so the boring part or the maybe the

abstract part that's going to maybe

confuse you is the definition of a

polynomial is ace to the N X to the n

plus a so the n minus 1 X to the n minus

1 plus dot dot dot there can be many

many terms in there going to ace to the

one X to the 1 plus ace to the 0 alright

and that still works on that so what

does all this mean well before I even

get to all what that means let's take a

look at our couple of constraints a sub

n cannot equal 0

you notice a sub n is going to be our

constraint that means we cannot just

have a 0 function of 0 that's not going

to be a polynomial function so a

synthetic cannot be 0 now that means any

one of those terms cannot just simply

equal 0 let's look at the next one n has

to be a non-negative integer so

therefore negative numbers or negative

exponents and rational exponents are not

going to be a part of polynomial

functions so let's take a look at some

examples and because I know this is very

abstract you're like I'm not really

getting that so let's take a look at

some examples how about f of X equals 3x

to the fourth minus 5x cubed plus 2x

squared plus X minus 5 all right you can

see right now my a sub n a sub n minus 1

all my my ace of my subscripts you can

see our my going to be my coefficients

of my variables and my n represents my

exponents all right and you can see as

we just kind of go into sending order my

negative 5 would represent my a sub 0

now let's take a look at

sorry that's what you get when you do

videos at out of school hope you don't

mind

so then the next one is f of X let's

look at another one that maybe not so

obvious f of X 3 f of X equals 3 X minus

3 and let's look at No 1 f of X equals 7

ok so these are all going to be

polynomial functions and the reason why

I'm even telling you to know about Ponyo

with functions because it's important

for you to understand even though these

don't directly look like this we can all

rewrite them in this type of forum where

we can relate our ace event okay is

going to be 3 here my n can be

re-written as an exponent of 1 and my

ace of 0 is going to be 3 or negative 3

in this case we usually just don't write

that though in this case we can write

this and say alright well I have an ace

of n which is going to be my ace of 0

which is going to equal 7 so now let's

look at some counter examples some

examples that are not polynomial

functions so we could say f of X equals

absolute value of 3x that is not a

function because it's an absolute value

function and it doesn't follow through

our definition f of X equals x2 negative

squared and f of X equals x to the 1/2

and f of X equals 0 so here are four

examples are non examples of polynomial

functions alright and we're actually

going to easier probably to explain why

an absolute value is not a polynomial

function once we get to the graphing

part which I will do this has a negative

power so it's not going to be a

polynomial function this has a rational

power so it's not going to be a

polynomial function and then obviously

this one says a sub n cannot equal 0

well if you just have 0 there it's not

going to be a polynomial function

so hopefully that little overview helped

you out and we will be using polynomial

functions for the rest of this course so

it's important that you have this basic

foundation of what a polynomial function

is thank you