in this lesson we're going to focus on

the rational zero theorem this theorem

helps us to list all the possible

rational zeros of a polynomial function

so it's very useful when solving

polynomial equations so here's an

example let's say that f of X is equal

to 1 X cubed plus 2x squared minus 5x

minus 6 and let's list all of the

possible rational zeros so we need to

divide P by Q and P is associated with

the constant term Q is associated with

the leading coefficient so factors of P

or factors of negative 6 include plus or

minus 1 plus or minus 2 plus or minus 3

and plus or minus 6

now factors of the leading coefficient 1

is just plus or minus 1 and any number

divided by 1 is itself so therefore the

possible rational zeros are 1 2 3 & 6 so

if we set this function equal to 0 then

the solutions to that equation will be

some of these numbers listed here so

let's go ahead and do that so let's say

that X cubed plus 2x squared minus 5x

minus 6 is equal to 0 let's calculate

the value of x in this problem now

notice that the degree of the polynomial

is 3 so that tells us that there's three

values of x these values could be real

numbers or imaginary numbers well let's

get those values now the purpose of the

rational zero theorem is for us to guess

the first is zero and then use that to

get the others using synthetic division

so we know the possible zeros 1 2 3 & 6

plus - so let's start with 1 let's see

if F of 1 is equal to zero so that's

going to be one race at a third power

plus two to

times 1 squared minus 5 times 1 minus 6

1 to the 3rd power is 1 and 1 squared is

1 times 2 is 2 5 times 1 is 5 and 1 plus

2 is 3 5 minus 6 is negative 11 and 3

minus 11 is negative 8 so notice that f

of 1 is not equal to 0 therefore 1 is

not a zero of a function of this

particular function now let's see if 2

is one of the zeros of this function so

this is going to be 2 to the 3rd power

plus 2 times 2 squared minus 5 times 2

minus 6 2 to the third power that's 2

times 2 times 2 that's 8 2 squared is 4

times 2 is 8

5 times 2 is 10 8 plus 8 is 16 negative

10 minus 6 is negative 16 and 16 minus

16 is 0 so f of 2 is equal to 0 that

means that x is equal to 2 so this is

one of the three zeros now let's use

this zero to get the other zeros and

this is when synthetic division will

come into play so the coefficients of

the polynomial are 1 2 negative 5 and

negative 6 so let's bring down to 1 2

times 1 is 2 and then we need to add 2

plus 2 is 4 and then 2 times 4 is 8 and

then negative 5 plus 8 is 3 and 2 times

3 is 6 and negative 6 plus 6 is 0 so

make sure this is a zero if not

something is wrong now notice that we

had X cube so this is going to be

associated with x squared so it's 1x

squared plus 4x plus 3 and that's equal

to zero now we need to factor this

expression so what two numbers multiply

to the constant term 3 and add up to the

middle coefficient

for this is gonna be three and one so

it's a factor it's going to be X plus

three times X plus one now if we set

each factor equal to zero X plus three

is equal to zero and X plus one is equal

to zero the two other answers will be

negative three and negative one and so

that's how you can use the rational zero

theorem to solve a polynomial equation

so first you need to list all the

possible zeros and then out of that

check to see which one will give you a

function value of zero so once you get

the first zero then you could use

synthetic division to get the other

zeros let's try another example

so let's find all the zeros of this

function let's say f of X is equal to X

cubed plus 8x squared plus 11x minus 20

so feel free to pause the video and try

so let's list the possible zeros factors

of the constant term P or factors of 20

are 1 2 4 5 10 and 20 and then for the

lien coefficient factors of 1 is just 1

so let's start with 1 let's see if F of

1 is equal to 0 so it's gonna be 1 to

the third power plus 8 times 1 squared

plus 11 times 1 minus 20 so 1 to the

third power is 1 1 squared is 1 times 8

and then this is going to be 11 1 plus 8

is 9 9 plus 11 is 20 and 20 minus 20 is

0 so 1 is 0 therefore we could say that

X is equal to 1 so that's the first

answer that we have

now let's use the entire division so the

coefficients are 1 8 11 and negative 20

so 1 times 1 is 1 8 plus 1 is 9 1 times

9 is 9 11 plus 9 is 21 times 20 is 20

negative 20 plus 20 is 0 so then this is

gonna be 1x squared plus 9x plus 20 now

let's factor it what are two numbers

that multiply to 20 but adds the middle

coefficient 9 so this is gonna be 4 & 5

4 times 5 is 20 4 plus 5 is 9 so it's a

factor it's going to be X plus 4 times X

plus 5 and let's set that equal to 0 so

therefore X will be equal to negative 4

and negative 5 so now we have three

zeros of the polynomial function 1

negative 5 and negative 4 let's work on

one more example so let's say that f of

X is X cubed minus 11x plus 6 the list

the possible zeros and find all zeros of

the function of f of X so factors of 6

are 1 2 3 & 6 and factors of the leading

coefficient are just 1 so let's start

with F of 1 so it's gonna be 1 to the

3rd minus 11 times 1 plus 6 so 1 minus

11 is negative 10 a negative 10 plus 6

is negative 4 so that is not 0 therefore

1 is not a zero of the function so let's

try 2 so it's going to be 2 to the 3rd

minus 11 times 2 plus 6 to the third is

8 11 times 2 is 22 and 8 minus 22 that's

negative 16 negative 16 plus 6 is

negative 10 and so that's not equal to 0

and that doesn't work either

actually my math is wrong eight minus

twenty-two is negative fourteen not

sixteen so let me just correct that at

negative 14 plus six that's negative

eight

so and that's still not equal to 0 so X

does not equal two

let's try three so this is going to be 3

to the third minus eleven times 3 plus 6

3 to the third is 27 11 times 3 is 33

now 27 minus 33 is negative 6 negative 6

plus 6 is 0 so therefore we found the

first 0 and that is X is equal to 3

so let's put a three and then this is

gonna be one now don't forget about the

0x squared which is between X cube and X

so we got to put a zero here and then

negative 11 and 6 so 3 times 1 is 3 3

times 3 is 9 negative 11 plus 9 is

negative 2 and 3 times negative 2 is

negative 6 so this is gonna be x squared

plus 3x minus 2 now we can't factor this

expression we can't find two numbers

that multiply to negative 2 and add to 3

so therefore we need to use the

quadratic equation so this is in the

form ax squared plus BX plus C so we can

see that a is equal to 1 B is equal to 3

and C is equal to negative 2 so here's

the quadratic formula it's a negative B

plus or minus the square root of b

squared minus 4ac divided by 2

I mean divided by 2a so it's gonna be a

negative B which is negative 3 plus or

minus B squared which is 3 squared so

that's 9 the minus 4 times a is 1 and C

is negative 2 divided by 2a which is 2

so negative 4 times negative 2 that's

plus 8 and 9 Plus 8 is 17 so this is

gonna be negative 3 plus or minus the

square root of 17 over 2 so the first

answer is going to be negative 3 plus

the square root of 17 over 2 and the

other answer is going to be negative 3

minus the square root of 17 over 2 so

these are the two other answers now

these are real numbers just with the

square root involved so we have X is

equal to 3 and negative 3 plus or minus

square root 17 over 2 but we still have

3 answers in total