Question

According to The Fundamental Theorem of Algebra, how many zeros does the function f(x) = 15x23 + 41x19 + 13x5– 10 have

Answer 1

The fundamental theorem of algebra states that a polynomial function of degree n will have n complex zeros


The degree of f(x) = 15x^23 + 41x^19 + 13x^5– 10 is 23 since that is the largest exponent

So there are 23 complex zeros (or just 23 zeros in general)

Answer 2

There's not much to do here but just memorize what the theorem is saying

Answer 3

ok lol i really hate mathhh!

Answer 4

well with more practice, it will get easier

Answer 5

haha if you say so... lets try another one okay
What are the zeros of the polynomial function: f(x) = x3 + 2x2 – 24x ?

Answer 6

Can you factor that at all?

Answer 7

do i just combine like terms or something.

Answer 8

What do x^3, 2x^2 and -24x all have in common?

Answer 9

they all have x?

Answer 10

good

Answer 11

so we can factor this out

f(x) = x^3 + 2x^2 – 24x

f(x) = x(x^2 + 2x – 24)

Answer 12

Now find two numbers that both multiply to -24 AND add to 2 at the same time

Answer 13

-12 and 2 ?

Answer 14

they multiply to -24, but do NOT add to 2

Answer 15

what else

Answer 16

ohhh wow my bad haha hold on ...

Answer 17

ok

Answer 18

-6 and 4

Answer 19

closer

Answer 20

but notice how -6+4 = -2

Answer 21

-4 and 6 ((:

Answer 22

better

Answer 23

so

x^2+2x-24

factors to

(x-4)(x+6)

Answer 24

so i could have just done the diamond problem? haha

Answer 25

so we can say

f(x) = x^3 + 2x^2 – 24x

f(x) = x(x^2 + 2x – 24)

f(x) = x(x-4)(x+6)

0 = x(x-4)(x+6)

x(x-4)(x+6) = 0

x = 0, x-4=0 or x+6 = 0

Answer 26

yes you could factored using the diamond method

Answer 27

could have factored*

Answer 28

okayyyyyy....lol

Answer 29

now solve for x in each equation

Answer 30

6,0,-4

Answer 31

oh wait, i went too fast lol

Answer 32

f(x) = x^3 + 2x^2 – 24x

f(x) = x(x^2 + 2x – 24)

f(x) = x(x-4)(x+6)

0 = x(x-4)(x+6)

x(x-4)(x+6) = 0

x = 0, x-4=0 or x+6 = 0

x = 0, x-4+4=0+4 or x+6-6 = 0-6

x = 0, x=4 or x = -6


So the zeros of f(x) = x^3 + 2x^2 – 24x are:

-6,0,4

Answer 33

yayyyyyyy i did it (((:

Answer 34

you were close, but it should be -6 instead of +6

Answer 35

so you got what I got?

Answer 36

yesss

Answer 37

alright great

Answer 38

What are the possible rational zeros of f(x) = x4 + 2x3 – 3x2 – 4x + 20 ?
what on earthhhh? i had subs my whole junior year in math):

Answer 39

List all the possible factors of 20

Answer 40

What are they

Answer 41

UHHHHHH....

Answer 42

2 AND 10

Answer 43

good, what else

Answer 44

20 = 2*10

Answer 45

5 and 4

Answer 46

good

20 = 5*4

Answer 47

anything else?

Answer 48

20 and 1

Answer 49

20 = 1*20

Answer 50

so we have

1*20,
2*10
4*5

Answer 51

Therefore, the factors of 20 are

20: 1, 2, 4, 5, 10, 20

Answer 52

Now divide each of these factors by the factors of the leading coefficient (which in this case, it's just 1)

1/1 = 1

2/1 = 2

4/1 = 4

5/1 = 5

10/1 = 10

20/1 = 20

So we get the list

1, 2, 4, 5, 10, 20

Now add on the negative of each number listed

1, 2, 4, 5, 10, 20
-1, -2, -4, -5, -10, -20

Answer 53

So all the possible rational roots are

1, 2, 4, 5, 10, 20
-1, -2, -4, -5, -10, -20

Answer 54

ohhh okay wow these seem hard but there really not if you know how to start them.

Answer 55

exactly, starting is the hardest part usually (for me anyways)

Answer 56

hahaha mee toooo!

Answer 57

that's why it helps to start at the end sometimes (ie start with what you want to get and see how to get there)

Answer 58

it's a bit odd, but it works sometimes

Answer 59

ive never thought about that. lol

Answer 60

it's a very useful trick that took me a while to grasp lol (i didn't learn it til recently either...and i still have trouble grasping it sometimes)