Descartes' Rule of Signs: 2x^4-7x^3+3x^2+8x-4
This equation shows that there would be either 3 positive roots, or 1 positive root, but my answer and the answer at the back of my textbook say that there are only two positive roots; 1/2, and 2.
The Descartes' Rule of Signs says that the number of positive roots is either equal to the number of variations in sign or is less than that by an even whole number. Why does that not apply to this equation?

Question

Descartes' Rule of Signs: 2x^4-7x^3+3x^2+8x-4
This equation shows that there would be either 3 positive roots, or 1 positive root, but my answer and the answer at the back of my textbook say that there are only two positive roots; 1/2, and 2.
The Descartes' Rule of Signs says that the number of positive roots is either equal to the number of variations in sign or is less than that by an even whole number. Why does that not apply to this equation?

amistre64 · Answer

what are the number of variations in your positive setup?

anonymous · Answer

I was thinking that it was 3 variations

amistre64 · Answer

2x^4 -7x^3 +3x^2 +8x -4
          |         |                 |
0       1         2               3

i count 3 as well

amistre64 · Answer

do you recall that a root can be repeated right?

anonymous · Answer

I was thinking about that also, but why would we have to subtract the number of variations by 2, instead of 1, if we were accounting for repeats?

amistre64 · Answer

the subtraction by 2 is to account for complex (nonreal) roots which always come in conjugate pairs

anonymous · Answer

I see. What would we do about the repeating roots, then?

amistre64 · Answer

repeating roots was just a thought, but it has 3 postivie roots http://www.wolframalpha.com/input/?i=0%3D2x^4-7x^3%2B3x^2%2B8x-4

amistre64 · Answer

does the question ask you to find the roots?  or just the number of them?

anonymous · Answer

It asks to find all the real roots.

amistre64 · Answer

and you are sure that your looking up the right answer key with the right question?  also, books do have errors in them

amistre64 · Answer

attaching a picture might help verify your cause :)

anonymous · Answer

Yep, it has the roots 1/2, 2, and -1. Doesn't the graph in the link have two positive roots and one negative root?  :o

amistre64 · Answer

i am so glad you are on top of these things ... so yeah, there is a double root.

so, it has 3 positive roots.  1/2, 2, 2  it is just that one of them occurs more than once.

anonymous · Answer

Okays, so when solving these kinds of problems, we should assume there is a double root if the answer isn't the correct number of variations?

amistre64 · Answer

you discover it either by working out the division ... or by the graph

anonymous · Answer

I see. Graphing...

amistre64 · Answer

as long as no errors occur in print, yeah, you can assume that one or more of the roots are a multiple root if we count 3 or 1, and only find '2' of them.

anonymous · Answer

Great, thanks!

amistre64 · Answer

your welcome