Question

can someone help me find the number of positive, negative and complex numbers of functions?

Answer 1

the first is f(x) = x^6 +x^5 +x^4 + 4x^3 - 12x^2 +12

Answer 2

in f(x), how many sign changes are there?

Answer 3

hmmm. 2

Answer 4

so there are at most 2 positive real roots

you could have 1 positive real root

or you could have no positive real roots

Answer 5

now find f(-x) and tell me what you get

Answer 6

f(-x)?

Answer 7

replace EVERY x with '-x' to get

f(x) = x^6 +x^5 +x^4 + 4x^3 - 12x^2 +12

f(-x) = (-x)^6 +(-x)^5 +(-x)^4 + 4(-x)^3 - 12(-x)^2 +12

I'll let you do the last step

Answer 8

im still confused sorry

Answer 9

-x is -1 ?

Answer 10

simplify (-x)^6 to get ______ (fill in the blank)

Answer 11

-6?

Answer 12

no

Answer 13

remember that exponents mean that you multiply the number by itself that many number of times

examples

7^3 = 7*7*7
8^5 = 8*8*8*8*8
9^14 = 9*9*9*9*9*9*9*9*9*9*9*9*9*9 ... note: there are 14 copies of 9 being multiplied

I'm sure you get the idea

Answer 14

yeah, but -x is not a number

Answer 15

is it?

Answer 16

yes it is, x stands in place of any number

Answer 17

what is (-x)^2 equal to

Answer 18

-2?

Answer 19

no 2

Answer 20

no

Answer 21

how are you getting rid of x?

Answer 22

x is 1 right

Answer 23

how do you know that x = 1?

Answer 24

i thought if there was no value in front it was 1

Answer 25

you're thinking of x = 1*x

Answer 26

but we don't know if x equals 1 or not

Answer 27

so what number is x?

Answer 28

we don't know

Answer 29

which is why we just leave it as x

Answer 30

so the function equals 12?

Answer 31

how are you getting that?

Answer 32

i don't have an idea

Answer 33

(-x)^2 = (-x)*(-x) = ???

Answer 34

-x^2

Answer 35

(-x)^2 = (-x)*(-x)

(-x)^2 = (-1*x)*(-1*x)

(-x)^2 = ???

Answer 36

i dont know

Answer 37

-1 * -1 = ???

Answer 38

2

Answer 39

no, I'm multiplying

Answer 40

1

Answer 41

yes, and x*x = ???

Answer 42

x

Answer 43

no

Answer 44

x^2

Answer 45

yes

Answer 46

that means we have

(-x)^2 = (-x)*(-x)

(-x)^2 = (-1*x)*(-1*x)

(-x)^2 = (-1*-1)*(x*x)

(-x)^2 = (1)*(x^2)

(-x)^2 = 1*x^2

(-x)^2 = x^2

Answer 47

now I want you to tell me what (-x)^3 is equal to

Answer 48

x^3

Answer 49

you sure?

Answer 50

yes

Answer 51

do the steps I did for (-x)^2, but instead, you'll have (-x)^3 = (-x)*(-x)*(-x)

Answer 52

(-x)^3 = (-x)*(-x)*(-x) = ???

Answer 53

-x^3

Answer 54

better

Answer 55

(-x)^4 = ???

Answer 56

x^4

Answer 57

(-x)^5 = ???

Answer 58

-x^5

Answer 59

(-x)^6 = ???

Answer 60

i get it if its odd its negative and even is positive

Answer 61

?

Answer 62

yeah that's a good way to remember it

Answer 63

f(x) = x^6 +x^5 +x^4 + 4x^3 - 12x^2 +12

f(-x) = (-x)^6 +(-x)^5 +(-x)^4 + 4(-x)^3 - 12(-x)^2 +12

f(-x) = ???

Answer 64

x^18+4?

Answer 65

what have we been doing before I posted my last message?

Answer 66

finding if they're negative or positive

Answer 67

so what does f(-x) = (-x)^6 +(-x)^5 +(-x)^4 + 4(-x)^3 - 12(-x)^2 +12 simplify to?

Answer 68

i don't know i get stuck after 4(-x)^3

Answer 69

replace (-x)^3 with what it's equal to

Answer 70

-x^3

Answer 71

4(-x)^3 turns into 4(-x^3)

Answer 72

so what does f(-x) = (-x)^6 +(-x)^5 +(-x)^4 + 4(-x)^3 - 12(-x)^2 +12 simplify to?

Answer 73

hmm

Answer 74

i don't know man

Answer 75

well we just went over what (-x)^6, (-x)^5, etc were equal to

Answer 76

tell me what you think f(-x) = (-x)^6 +(-x)^5 +(-x)^4 + 4(-x)^3 - 12(-x)^2 +12 simplifies to?

Answer 77

x^18+16

Answer 78

how are you getting 18?

Answer 79

12 + 4

Answer 80

replace (-x)^6 with what it's equal to and you get what?

Answer 81

x^6

Answer 82

f(-x) = (-x)^6 +(-x)^5 +(-x)^4 + 4(-x)^3 - 12(-x)^2

f(-x) = x^6 +(-x)^5 +(-x)^4 + 4(-x)^3 - 12(-x)^2

Now do the same for (-x)^5

Answer 83

-x^5

Answer 84

f(-x) = (-x)^6 +(-x)^5 +(-x)^4 + 4(-x)^3 - 12(-x)^2

f(-x) = x^6 +(-x)^5 +(-x)^4 + 4(-x)^3 - 12(-x)^2

f(-x) = x^6 + (-x^5) +(-x)^4 + 4(-x)^3 - 12(-x)^2

f(-x) = x^6 - x^5 +(-x)^4 + 4(-x)^3 - 12(-x)^2


Now do the same for (-x)^4

Answer 85

x^4

Answer 86

f(-x) = (-x)^6 +(-x)^5 +(-x)^4 + 4(-x)^3 - 12(-x)^2

f(-x) = x^6 +(-x)^5 +(-x)^4 + 4(-x)^3 - 12(-x)^2

f(-x) = x^6 + (-x^5) +(-x)^4 + 4(-x)^3 - 12(-x)^2

f(-x) = x^6 - x^5 +(-x)^4 + 4(-x)^3 - 12(-x)^2

f(-x) = x^6 - x^5 + x^4 + 4(-x)^3 - 12(-x)^2


Now do the same for (-x)^3. See how this works?

Answer 87

-x^3

Answer 88

finish up and tell me what you get

Answer 89

i don't know how to simplify this

Answer 90

why do I get the feeling you just don't want to work

Answer 91

? im trying

Answer 92

x^5 + 8x?

Answer 93

4(-x)^3 turns into ???

Answer 94

-4x^3

Answer 95

f(-x) = (-x)^6 +(-x)^5 +(-x)^4 + 4(-x)^3 - 12(-x)^2

f(-x) = x^6 +(-x)^5 +(-x)^4 + 4(-x)^3 - 12(-x)^2

f(-x) = x^6 + (-x^5) +(-x)^4 + 4(-x)^3 - 12(-x)^2

f(-x) = x^6 - x^5 +(-x)^4 + 4(-x)^3 - 12(-x)^2

f(-x) = x^6 - x^5 + x^4 + 4(-x)^3 - 12(-x)^2

f(-x) = x^6 - x^5 + x^4 + (-4x^3) - 12(-x)^2

f(-x) = x^6 - x^5 + x^4 - 4x^3 - 12(-x)^2

I'll let you do the last step

Answer 96

12x^2

Answer 97

f(-x) = (-x)^6 +(-x)^5 +(-x)^4 + 4(-x)^3 - 12(-x)^2

f(-x) = x^6 +(-x)^5 +(-x)^4 + 4(-x)^3 - 12(-x)^2

f(-x) = x^6 + (-x^5) +(-x)^4 + 4(-x)^3 - 12(-x)^2

f(-x) = x^6 - x^5 +(-x)^4 + 4(-x)^3 - 12(-x)^2

f(-x) = x^6 - x^5 + x^4 + 4(-x)^3 - 12(-x)^2

f(-x) = x^6 - x^5 + x^4 + (-4x^3) - 12(-x)^2

f(-x) = x^6 - x^5 + x^4 - 4x^3 - 12(-x)^2

f(-x) = x^6 - x^5 + x^4 - 4x^3 - 12x^2

Answer 98

what do i do when i subtract x^6 and x^5. subtract exponents?