There are plenty of sites available to help you understand the rule of signs. How much are you able to do? Where are you struggling? I think the questions are quite straightforward.
Sometimes finding the factors of the polynomial function can be very time consuming. The french mathematician rene descartes discovered that if a polynomial function f(x) has real coefficients and is written in decreasing degree, then the number of positive real zeros that the function has is the same as the number of sign changes in f(x) as you follow it from term to term or is less than that by multiple of two. The number of negative real seros that the function has is the same as the number of sign changes in the polynomial functon f(-x) or is less that that by a multiple of two. This is known as Descartes Rule of Signs.
Consider the polynomial function: f(x) = x^5 - 5x^4 + 3x^3 + 17x^2 - 28x + 12
a) how many sign changes are there in f(x)?
B)According to Descartes rule of signs how many positive real zeroes might f(x) have?
c)determine f(-x)
d)how many sign changes are there in f(-x)
e)According to descartes rule of signs how many negative real zeroes might f(x) have?
f) based on your conclusions in (b) and (e) use synthetic division to factor f(x).
G) draw a sketch of f(x)
a) How many sign changes?
I'll colour them:
f(x) = + x^5 - 5x^4 + 3x^3 + 17x^2 - 28x + 12
That is four sign changes.
b) How many positive real roots?
There could be up to four. That is what is significant about the rule of signs. There could be two, or there could be none, but the maximum is four.
Can you try the other parts?