Question 202318: given x^3 - 4x^2 + 2x + 1= 0 (a)how many possible positive roots are there? (b)what are the possible rational roots? (c)using synthetic substitution, which of the possible rational roots is actually a root of the equation? (d)find the irrational roots of the equation
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website!
For the first question you need Descarte's Rule of Signs. The number of positive roots is equal number of changes of sign on the coefficients or is less than that by any multiple of 2. In this equation, the lead coefficient is positive, the next is negative. That is one change of sign. The next coefficient is positive, that is a second change of sign, and the final coefficient is also positive, so in total we have two sign changes meaning the maximum number of positive roots is 2 (note that there could also be zero positive roots).
For the second question you need the Rational Root Theorem. If a polynomial function, written in descending order, has integer coefficients, then any rational zero must be of the form  , where  is a factor of the constant term and  is a factor of the leading coefficient. For your equation, the leading coefficient is 1 and the constant term is 1, so the only possible rational roots are 
Synthetic division is a big pain in the neck to render properly on this system, so I'll just tell you can review the process here:
http://www.purplemath.com/modules/synthdiv.htm
And that +1 is a root of the equation.
The quotient of the synthetic division is 1 -3 -1, which translates to:
which can easily be solved using the quadratic formula to determine the two irrational roots that remain.
John
|
|
|
