Question

How many real roots does the polynomial 2x^5+8x-7 have?

Answer 1

Hello

Answer 2

I know the answer, but I'm looking for a quick method to solve these kinds of problems.

Answer 3

Haha...... this is wierd I cant beleive Im helping someone who is smarter than me

Answer 4

use a graphing calculator

Answer 5

I took the derivative and saw that it was always positive, so I knew it had one root. But is there some way in general to tell how many roots a polynomial has?

I can't use a graphing calculator, this is a question from the math gre subject test.

Answer 6

Well the degree tells us that there will be at most 5 real roots, and then you can find the factors etc haha...idk if that helps you though

Answer 7

Yeah, I know it can have at most 5 roots, but is there some sort of secret trick to easily see if we can break it down to 4, 3, 2, or 1?

Answer 8

For an odd degree polynomial, I guess it always has to have at least one root, and I guess I should check to see where the local mins and maxes are, if they are at x=0 then there will be an even number of roots I suppose hmm I don't know maybe I'm over thinking this lol.

Answer 9

Group terms and factor is the fastest way I can see, mhm...

Answer 10

Hey, I just read something, Descartes's Rule might help you out

Answer 11

why dont we use descartes rule from precalculus ?

Answer 12

lol

Answer 13

http://sepwww.stanford.edu/oldsep/stew/descartes.pdf

proof is bit hard though

Answer 14

@ganeshie8 Could you really fast show me how you would use it for this problem just to give me an idea of how simple/hard the method is and what it looks like?

Answer 15

Let \(f(x) = 2x^5 \color{Red}{+} 8x\color{Red}{-}7\)
number of sign changes = 1, so there will be exactly one positive real root

\(f(-x) = -2x^5 -8x - 7\)
number of sign changes = 0, so there wont be any negative real roots

Answer 16

Hey kai, so from what I understand, you look at the polynomial in descending order, and pay attention to how many times the sign changes from term to term, this tells you the maximum number of positive roots, kind of like a pattern. So how many times do you see changes of sign in your polynomial? Which will tell you the amount of positive solutions.

Answer 17

Haha, yeah what ganeshie wrote! :P

Answer 18

Thanks everyone! =)

Answer 19

@Kainui I don't think that an odd degree polynomial can have an even number of or 0 real roots.