according to descartes" rule of signs, how many possible negative real numbers could this polynomial have?
f(x)=x^4+x^3-5x^2+x-6

Question

according to descartes" rule of signs, how many possible negative real numbers could this polynomial have?
f(x)=x^4+x^3-5x^2+x-6

nory · Answer

How many changes of sign does this polynomial have?

nory · Answer

Oops, wait--I was thinking of the positive.

anonymous · Answer

I dont understand :(

nory · Answer

Sorry.
1. Multiply the polynomial by -1. What do you get?

anonymous · Answer

-x^4-x^3+5x^2-x+6

anonymous · Answer

i think

nory · Answer

Looks good.
Now, count the number of changes of sign in the polynomial. Do you know what that means? (If not, I can explain.)

anonymous · Answer

yeah explain please

nory · Answer

Oh wait, I got something wrong. So sorry! :(
For the original polynomial function f(x), find f(-1). so evaluate the polynomial at x = -1.

nory · Answer

While you do that, I'll tell you what a sign change is:

anonymous · Answer

-12?

nory · Answer

It's when a term goes from positive to negative.
ex. x^2 - 3. This has one sign change. The first term has a positive coefficient, but the second term has a negative coefficient. Do you see?

anonymous · Answer

yeah

nory · Answer

wait...not f(-1), f(-x).
(Sorry...that's my final revision.)
Then what do you get?

anonymous · Answer

idk :(

nory · Answer

I know, it's really weird.
Basically, make every other coefficient negative.
ex. If f(x) = x^2 - 4x + 1, what is f(-x)?
Every other term, multiply the coefficient by -1.
So you get x^2 +4x + 1.
Does that make sense? I didn't explain it very well.

anonymous · Answer

what woud the answer be?

nory · Answer

x^4 - x^3 - 5x^2 - x - 6

nory · Answer

There is 1 change of sign in that, so the answer is...wait, I think something went wrong here.

nory · Answer

I have to go.