Polynomial Functions | Don't Memorise


what is the polynomial function it is a

function of the form f of X is equal to

n multiplied by X raise to n plus a and

minus 1 multiplied by X raise to n minus

1 and so on up to a 0 we will begin by

making a few observations so how many

variables do we see here we see only one

variable X so we can say f of X is a

function in X we notice that X terms are

raised to unique powers so why do we use

the word unique this is because no two

terms of X at the same power yes we

notice that every X term is raised to a

unique power but what is the power of X

in the term a zero to answer this

question we need to understand that a

zero is nothing but a zero multiplied by

X raise to 0 so the power of the X term

is zero so we can say that the X terms

have non-negative integral powers so X

can be raised to positive integers or 0

also every term of X has a coefficient

the coefficients are a 0 a 1 a 2 and so

on up to a n these are all real

coefficients now in what order do we

write the terms should the terms be

written and increasing or decreasing

powers of X both the ways are

mathematically correct since they result

in the same function in our video we

have chosen to write the function in the

order of decreasing powers of X writing

the terms in descending order of power

is called the standard form now what's

the degree of a polynomial function can

we say that the degree of a function is

n this happens when the highest power of

X is N and the coefficient of x raise to

n a n is not equal to 0

what happens when a n is equal to zero

then the term a n multiplied by X raise

to n would be zero making the degree of

the polynomial n minus 1 so we know what

the degree of a polynomial is now we

look closely at the terms to understand

the anatomy of our polynomial function

here is our polynomial function of X a n

is the leading coefficient this is

because it's the coefficient of the X

term with the highest power n can you

guess what the terms a 0 a 1 times X and

times x squared are called they are

called the constant term the linear term

and the quadratic term respectively

this is because the power of X in each

of these terms is 0 1 & 2 respectively

now let's test our understanding to see

if the following functions are

polynomial functions or not if F in

brackets X equal to 5 a polynomial

function yes it's a polynomial function

of degree 0 this is because 5 can be

written as 5 x x raise to 0 here the

highest degree of the function is 0 what

about these two functions f of X equal

to 2 multiplied by X raise to 4 over 5

plus 2 and the function f of X is equal

to 6 over x squared

are they polynomial functions or not

they are both not polynomial functions

this is because the X term in f of X is

raised to a fractional power for it to

be a polynomial the power of X can only

be unknown negative integer similarly in

the second function the X term in press

of X is equal to a negative power hence

they both aren't polynomial functions

how about the function f of X is equal

to X raise to 4 plus X is this a

polynomial function we notice that the X

terms with powers 3/2 and 0 are missing

can we say that it's a polynomial

function of degree 4 absolutely a

polynomial function can have some X

terms missing so this discussion gave us

an overview about the

basic concepts of polynomial functions

in the next video we will learn about

rational functions