Given a Polynomial Function Find All of the Zeros

oh I forgot too much thing so if you

guys don't mind I'm going to kind of go

through this I'm gonna go through the

rational zero test and I'm gonna go

through the carts realest signs fairly

quickly but the reason being is because

there's more to this problem that I want

to get to because you guys will have

problems that will say hey what's the

rational zero test you know what's the

possible rational zeros they'll have

problems that hey what's the real zeros

but really all that kind of stuff is you

know why do we even care to know it why

why are we going to be using it so let's

do it first do the P over Q so I just

say P Q so P is 24 so we have 24 times

1/12 times 2 8 times 3 and 6 times 4 yes


so if I was going to do P over Q

remember it's the factors plus or minus

P over Q well my Q the coefficient of Q

is 1 so we know the only factors Justin

of 1 or 1 and 1 right so therefore it's

going to be 24 12 8 6 4 3 2 1 all over 1

so when doing my factors you can say

well all the possible rational zeros P

over Q is just going to be plus or minus

now you can write plus or minus for

every single term if you want to or you

can also just put plus or minus outside

of parenthesis and say plus or minus is

going to be distributed to all of them

and just write that as 24 12 8 6 4 3 2 1

okay very good okay so now let's go

ahead and determine

now let's go ahead and determine the

positive real and negative zeros so when

I take a look at the positive I'll just

take all this rich take out the signs

goes to positive to negative to positive

to positive to negative right all I do

is I took the sign of each one of my

monomials and I brought it down

therefore you can see I have three of

them right so three minus an even number

it one even number is 2 so 3 minus 2

would be 1 so therefore I have 3 or 1

positive real zero correct okay now

let's go and look at the negative now

I'm gonna kind of do this in my head

because I already showed you guys what

to do yes you yes you can keep on

subtracting so let's say I gave you 10

let's say there's 10 alternating signs

then the number of possible real zeros

would be 10 8 6 4 2 & 0 okay so now

let's go and look at the negative now

remember the negative you need to

evaluate for F of negative x to keep

this video a little short I will I'll do

then I'll do the signs out here so

that's going to be positive that will be

positive that will be positive that will

be negative and that will be negative I

did this in my head okay you guys can

check my work but you can see that

there's going to be one alternating sign

all right so therefore we'll have one

real negative one real negative zero

okay so so what we have right now ladies

and gentleman I'm asking you to find the

zeros okay I ask you to find the zeros

let's say you don't have a graphing

calculator all right you're not allowed

to use your phone for google so I say

you're gonna have three positives and

one zero or one of them is going to be

negative and if it's rational here are

all your zeros so how do we determine if

something is if we're given a zero how

can we determine if it's a zero of a

polynomial it kind of goes back up to

what my first problem I did today what

can we do we can do how cool we can

determine if I give you a zero if I say

a zero if I say zero equals one how do

you know that 1 is a zero what can you

do what process can we do yes you can

plug it into the equation right and if

you plug it into the equation and you

get zero then you know it is a zero of

the function right very good however if

I get up to like 24 I'm probably gonna

want to plug 24 into this it's just give

me a lot of extra math right and then

plus what do I do after that how is that

else gonna help me so let's say you plug

it in and you get a zero well alright so

that tells you it's a zero but what do

you do from there so what else can you

determine the zero because plugging it

is a great way to determine it but

there's another way that we turned by

using the remainder theorem right the

remainder theorem was plugging it in but

then we could also check it by what

process I did it today synthetic

division and what's nice about synthetic

division it's the exact same thing as

what you said but when you do synthetic

division what do you get your answer you

get your answer is what we call the what

quotient and that quotient is what of

the polynomial a factor and a fat from a

factor we can find what the remaining

zeros right so here's the problem though

I don't know which one is a zero or not

right so guess what if you don't know

what the zeros are you're just gonna

have to check

so if you don't know what the zeros are

guess what ladies and gentlemen you're

just gonna have to guess and check why

start with the heart why start with 24

let's do the easy 1 1 once you say 1 is

probably pretty simple to start and then

if one doesn't work we'll do negative 1

all right so let's do 1 bring down the 1

1 times 1 is 1 negative 6 negative 6 4 4

18 so that doesn't work right so you go

back to drawing board say ok that didn't

work let's do negative 1 so you bring

down so negative 1 doesn't work so what

do you do negative 1 doesn't work I've

moved it to right and then do it ball 4

2 and negative 2 then I've moved to 3

and try 3 and negative 3 now I use

graphing technology and I know that 3

works so if you didn't use graphing

technology you'd probably a couple

minutes behind me because you'd be doing

synthetic division for all of these

until you got up to 3 so I do 3 bring

down the 1 1 times 3 is 3 negative 4

negative 12 negative 2 negative 6

negative 8 no no no it's a positive 8

positive 8

yes ok so now that's nice so now I have

a polynomial

remember remainder constant linear

quadratic and cubic whoo but now you

guys remember when I gave you that one 0

and I applied synthetic division and I

said hey once you have 0 then you can

keep on doing it right ok so we know

that 1 is not a zero we know that's 2 or

negative 2 is now to 0 now we know 3 is

a 0 but you can either try to factor

this so you can say you know is this

quadratic or not quadratic but is this


is that factorable and you could factor

it by what grouping right

okay now you could factor that and we

all determine so here I would be able to

factor out yeah you can go in factor

that by grouping wouldn't be a problem

however if you use your graphing

technology you would also notice that 4

is also a zero so rather than factoring

it I already know I use graphing

technology so I know that 4 is also a

zero and I'll show you guys how to

determine at with your graphing


so rather than factoring because you

know maybe I'm just not good at

factoring so I don't want to do it so

I'll go and factor it again by doing

synthetic division I bring down the 1 1

times 4 is 4 negative 3 negative 12

negative 2 negative 8 positive 6 24 0

now I have a 0 constant linear did I

pick the wrong one I rewrote it ha ha ha

all right big mistake you don't want to

make sure you do this if you guys notice

if I rewrite the factors same thing I'm

just going to get another factor I'm not

simplifying at all when you refactor you

have to take your previous quotient and

then factor that down further my mistake

sorry about that because you guys notice

when I did that again I get in getting a

pie I'm getting a factor it to the third

power so take 1 negative 4 negative 2

and 8 bring down bring down the 1 1

times 4 is 4 0 0 times 4 is 0 negative 2

negative 8 all right therefore that

becomes 0 now you guys see you have a

remainder constant linear quadratic so

you have x squared minus 2 that is your

factor right and can you solve the other

two zeros by the using that factor yes

so let's go into we set it equal to 0

I'll do it over here x squared minus 2

equals 0 x squared equals 2 square root

square root x equals plus or minus the

square root of 2 now is the square root

of 2 is that a rational number or is

that a irrational number right so that's

why it's

- is not here because it's not rational

it's irrational now let's count the

number of zeros we have the one zero

which is three right so we know those

are zeros and also x equals three how

many positive zeros do I have to

that's issue so I'm missing a zero three

no oh I'm sorry and I forgot to write

down that one we said four is the zero

as well right so how many positive zeros

do I have three how many negative zeros

don't have one and are those real just

because it's irrational still real right

so you guys can see it follows this case

now however let's pretend I only had one

positive if I only have one positive and

I only had one negative that means I

still am missing two more zeros if

they're not real then they have to be

imaginary okay so in this problem that

we have all real zeros which is cool but

you should know that win by using the

cultural signs if you don't have enough

to count up the real that means the

other ones are imaginary yes yes you

will have a problem just like this using

graphing that you'll either have to do

synthetic division until you find the

zeros or use graphing technology yes

yeah I mean you can start from the

highest but yes I would recommend

starting from the lowest yeah then you

fight once you've once you figure out

one zero then you take the quotient and

then you keep on working the synthetic

division so you find another zero ok now

I'm gonna talk to you though about your

graphing calculator how you don't have

to do all that legwork