Question

Help? Using complete sentences, describe how you would analyze the zeros of the polynomial function f(x) = –3x^5 – 8x^4 +25x^3 – 8x^2 +x – 19 using Descartes’ Rule of Signs.

Answer 1

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

Answer 2

I don't know?

Answer 3

notice that going from -8x^4 to +25x^3, there's a sign change from negative to positive

Answer 4

do you see this?

Answer 5

Yes

Answer 6

ok there's another from +25x^3 to -8x^2

Answer 7

what's another one?

Answer 8

+x to -19?

Answer 9

there's one more

Answer 10

from -8x^2 to +x

Answer 11

so there are 4 sign changes total in f(x)

Answer 12

this means that there are at most 4 positive real roots

Answer 13

-19 to +x?

Answer 14

no that's the same sign change (just in reverse)

Answer 15

4 sign changes=4 positive real roots

Answer 16

at most 4 (there could be 0, 1, 2, 3, or 4 positive real roots)

Answer 17

4 is the maximum

Answer 18

now we must find f(-x)

Answer 19

I'm still really confuzzled...

Answer 20

where at?

Answer 21

All of it

Answer 22

the rule is

if f(x) has n sign changes, then there are AT MOST n positive real roots

Answer 23

if n = 4, then

if f(x) has 4 sign changes, then there are AT MOST 4 positive real roots

Answer 24

Okay

Answer 25

making more sense?

Answer 26

...

Answer 27

yes? no?

Answer 28

I don't know how to formulate an answer from you telling me this...

Answer 29

you don't need to find the actual roots

you just need to be able to count the possible number and type

Answer 30

more specifically, how many positive real roots you could have (in this case, at most 4)

you don't need to find the 4 or find out how many actual positive roots there are

Answer 31

But I need a definite answer, right?

Answer 32

I can't just say "maybe 4"?

Answer 33

well that's part of the answer, we still have to find f(-x)

Answer 34

Okay, and how would we go about that ? :)

Answer 35

start with f(x)

then replace each x with -x and simplify

Answer 36

Could you give me an example?

Answer 37

start with this

f(x) = –3x^5 – 8x^4 +25x^3 – 8x^2 +x – 19

then replace each 'x' with '-x' to get

f(-x) = –3(-x)^5 – 8(-x)^4 +25(-x)^3 – 8(-x)^2 +(-x) – 19

then simplify to get

???

Answer 38

does that help?

Answer 39

Yes, it does :)

Answer 40

ok what do you get

Answer 41

when you simplify

Answer 42

3x^5=f(-x)+x(x(x8x(+25)+8)+1)+19??

Answer 43

f(-x) = –3(-x)^5 – 8(-x)^4 +25(-x)^3 – 8(-x)^2 +(-x) – 19

would simplify to

f(-x) = 3x^5 – 8x^4 - 25x^3 – 8x^2 - x – 19

Answer 44

now count the sign changes in

f(-x) = 3x^5 – 8x^4 - 25x^3 – 8x^2 - x – 19

and tell me how many you got

Answer 45

No sign changes?

Answer 46

3x^5 is the same as +3x^5

Answer 47

so there's only one sign change from +3x^5 to -8x^4

Answer 48

so this means that there is at most 1 negative real root

Answer 49

Oh, I was just looking at it wrong.. Sorry :(

Answer 50

no worries

Answer 51

So far we found that there are at most 4 positive real roots and at most 1 negative real root

Answer 52

I see

Answer 53

so one possibility is that there are 4+1 =5 real roots (4 positive, 1 negative)

BUT

this is not the only possibility

Answer 54

we could have 3 positive real roots and 1 negative real root to have 3+1 = 4 real roots total

but the problem with this scenario is that you would only have 5-1 = 1 complex root...when complex roots should come in pairs

Answer 55

so that scenario of 3 positive real roots and 1 negative real root just isn't possible

Answer 56

Okay
So don't worry about the last scenario?

Answer 57

anyways, there are a ton of possibilities

the good news is that you can sum them all up by saying



there are at most 4 positive real roots (you could have 0, 1, 2, 3, or 4 positive real roots)
there is at most 1 negative real root (you could have 0 or 1 negative real roots)
there are at most 5 complex roots (if there are no real roots at all)

Answer 58

in short

there are at most 4 positive real roots
there is at most 1 negative real root
there are at most 5 complex roots

Answer 59

I got it! :D

Answer 60

ok great

Answer 61

Thanks!

Answer 62

we can stop there because we don't need to find the actual roots or the counts of the types or roots...just the max of all of the possible types of roots

Answer 63

sure thing