Question

Help with descartes rules of signs

Answer 1

g(x) = x^2 − 5x

Answer 2

this has one change if i am correct

Answer 3

@ganeshie8

Answer 4

yes so one positive root for sure ?

Answer 5

what about g(-x) ?

Answer 6

for -x also 1

Answer 7

because x^2 doesn't change

Answer 8

but 5x does

Answer 9

g(-x) = (-x)^2 − 5(-x)
= x^2 + 5x

Answer 10

there are no sign changes,
so there won't be any negative real roots

Answer 11

wait i lied

Answer 12

yah for some odd reason i still pictured - 5x in my mind and this is why it is good to write out things aha

Answer 13

so I believe for this it would only be 1 positive zero then?

Answer 14

yes as you can see
x^2-5x = x(x-5)

so x = 0 and x=5 are the zeroes

Answer 15

Notice that the positive zero is 5

Answer 16

0 is not positive, not negative

Answer 17

alright. Could you do two more with me though just to make sure i get it aha

Answer 18

ok sure

Answer 19

Use Descartes' Rule of Signs to determine the number of positive real zeros of the following function. f(x) = x^3 − 6x^2 + 11x − 6
3 changes

and for -x
-x^3 - 6x^2 -11x -6 so 0 change for negative

Answer 20

f(x) is sufficient to determine positive real zeroes

Answer 21

f(x) = x^3 − 6x^2 + 11x − 6
3 changes

Answer 22

so there will be `3` or `1` positive real zeroes

Answer 23

because the complex zeroes in pairs...

Answer 24

wait why or 1 though?

Answer 25

thats how descartes rule of signs works

Answer 26

oks

Answer 27

if you have `n` sign changes in f(x) :

the positive real zeroes can be `n` or `n-2` or `n-4` or ...

Answer 28

for example

if u have `5` sign changes in f(x) :

the positive real zeroes can be `5` or `3` or `1`

Answer 29

ohhhhh oks that makes sense now

Answer 30

good

Answer 31

Use Descartes' Rule of Signs to determine the maximum number of positive and negative real zeros of the following function:

f(x) = 2x^4 − 15x^3 + 23x^2 + 15x − 25

Answer 32

3 changes

and for -x
2x^4 +15x^3 +23x^2 -15x -25
1 change

Answer 33

So 1 negative zero for sure

Answer 34

since the degree is 4, there can be atmost 4 different zeroes only

Answer 35

1 negative zero
3 positive zeroes / 0 complex zeroes


OR

1 negative zero
1 positive zeroes / 2 complex zeroes

Answer 36

it could be like that ^^

Answer 37

what is a complex zero exactly aha

Answer 38

they are zeroes of form : a + ib

Answer 39

real zeroes are of form : a

Answer 40

ahhh i see

Answer 41

Thank you for your help once again friend

Answer 42

np :)