Question

Use the Descartes's Rule of Signs to determine the possible numbers of positive and negative zeros of the function:

f(x)= -5x^3+X^2-X+5

Answer 1

How many times did the expression changed sign?

Answer 2

3

Answer 3

i think

Answer 4

There is then a MAXIMUM of 3 positive roots, but it could also be 1.

Answer 5

why 1?

Answer 6

Now flip the sign of the coefficients of the odd powers.
How many times does the new expression change sign?

Answer 7

Is that 5X^3+X^2+X+5

Answer 8

The number of sign changes is the maximum number of positive roots, including complex.
If there are two complex roots (as is the case), then there is only one (real) positive root.

Answer 9

ok

Answer 10

so the negative right converted right?

Answer 11

Yes, the number of sign changes of the new expression (i.e. replace x by -x) gives the MAXIMUM number of negative roots.

Answer 12

So for this one there is none for the negative and 1 for the positive

Answer 13

so how do you know there is one and not 3 for the positive?

Answer 14

Using the Descartes rule of signs, we do know that there is no negative root.
But we do not know if there is ONE or THREE positive roots.

Answer 15

ok so the answer for positive is 1 and 3
and negative is 0

Answer 16

You can find that out by factoring the expression.
A cubic has at least one real root, and we know that it is positive in this case.

Answer 17

correct.

Answer 18

one other question:

Answer 19

Use the Descartes's Rule of Signs to determine the possible numbers of positive and negative zeros of the function:

f(x)=3X^2+2X^2+x+3

Answer 20

how does this work then?

Answer 21

the positive looks like there is none and same with the negative?

Answer 22

Now, let's see.
(1) The expression as is has no change in sign, so what do you conclude?
(2) flip the sign of the odd powers to give -3X^3+2X^2-x+3, sign changed 3 times, what do you conclude?

Answer 23

wait why is 3X^2 flipped?

Answer 24

I supposed you have a typo!
It's 3x^3, ... or not?

Answer 25

let me see...

Answer 26

oh srry your right

Answer 27

Use the Descartes's Rule of Signs to determine the possible numbers of positive and negative zeros of the function:

f(x)=3X^3+2X^2+x+3

Answer 28

ok so the positive is none
then the negative is -3X^3+3X^2-X+3, so 2 and 1

Answer 29

No positive is correct.
Can you repeat the conclusion for negative?

Answer 30

i don't understand

Answer 31

of so negative is 3 and 1

Answer 32

The number of possible roots always differ by an even number because complex roots, if any, come in pairs.
1 and 2 will not be correct.
1 and 3 would be.

Answer 33

oh ok

Answer 34

Cool!

Answer 35

cool?

Answer 36

Perhaps more exactly, 1 OR 3.

Answer 37

ok thanks that makes sense

Answer 38

You're welcome! :)