# Polynomial and Rational functions - Descartes Rule Of Signs

• Jun 5th 2011, 04:18 PM
smartboy
Polynomial and Rational functions - Descartes Rule Of Signs
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)
• Jun 5th 2011, 04:50 PM
Quacky
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.
• Jun 5th 2011, 04:54 PM
smartboy
im struggling at understanding it i have no clue how to get the zeroes, because arent they in the denominator there is no denomintor??
• Jun 5th 2011, 05:02 PM
Quacky
\$\displaystyle f(x) = x^5 - 5x^4 + 3x^3 + 17x^2 - 28x + 12\$

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?
• Jun 5th 2011, 05:21 PM
smartboy
why did u colour the + signs i dont understand why?
• Jun 6th 2011, 01:18 AM
HallsofIvy
Because the problem asked for the number of sign changes. By coloring the "+" signs differently from the "-" signs, Quacky was trying to help you count the number of changes. Can you not at least count them?