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

I don't know?

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

do you see this?

Yes

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

what's another one?

+x to -19?

there's one more

from -8x^2 to +x

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

this means that there are at most 4 positive real roots

-19 to +x?

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

4 sign changes=4 positive real roots

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

4 is the maximum

now we must find f(-x)

I'm still really confuzzled...

where at?

All of it

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

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

Okay

making more sense?

...

yes? no?

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

But I need a definite answer, right?

I can't just say "maybe 4"?

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

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

start with f(x)
then replace each x with -x and simplify

Could you give me an example?

does that help?

Yes, it does :)

ok what do you get

when you simplify

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

No sign changes?

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

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

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

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

no worries

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

I see

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

Okay
So don't worry about the last scenario?

I got it! :D

ok great

Thanks!

sure thing