Question

How many negative roots can this polynomial function possibly have?
f(x) = 5x^4 – 7x^3 – x^2 – 3x – 2

A.1
B.5 or 3 or 1
C.3 or 1
D.4 or 2 or 0
E. 2 or 0

Answer 1

f(x)
+---- Must have one Positive Real Root
f(-x)
++-+- Has 3 or 1 Negative Real Root.

I would have throught the problem statement more technically correct had it sair "Real".

Answer 2

so its c?

Answer 3

Why do you doubt? Do you know the rule?

Answer 4

no

Answer 5

One premise is this, with Real Coefficients, Complex Roots must come in pairs. This will be important later.

Count the sign changes in f(x) +---- It changes from + to - once and stays there. This is ONE sign change. Since this was f(x), there MUST BE a Positive Real Root.


Count the sign changes in f(-x) ++-+- It changes from + to - , then back to + and finally back to -. There are THREE sign changes. Since this was f(-x), there CAN BE as many as THREE (3) Negative Real Roots. If there are not 3, there must be fewer buy some factor of 2, namely, in this case, one (1).

Answer 6

It is called "Descartes' Rule of Signs".

Answer 7

ahh is see thanks