Question

Use Descarte's Rule Of Signs to determine the possible numbers of positive and negative real zeroes of f(x)=-5x^3+9x^2-4x+2

Answer 1

Descartes' Rule of Signs states that:
1. The number of positive real zeros is either equal to the number of changes in the sign in P(x), or is less than that by an even whole number.
2. The number of negative real zeros is either equal to the number of changes in the sign in P(-x), or less than that by an even whole number.

P(x)=-5x^3+9x^2-4x+2
The 1st term is -.
The 2nd term is +.
The 3rd term is -.
The 4th term is +.
There are 3 changes in the signs.

Now plug in x=-x
P(-x)=-5(-x)^3+9(-x)^2-4(-x)+2
5x^3+9x^2+4x+2
All 4 terms are +.
There are 0 changes in the signs.

This means that this polynomial has either 3 or 1 positive zeros, and 0 negative zeros, making a total of 3 or 1 real zeros.