algebra 2 problem

Question

I have no idea what the heck Descartes’ rule of signs is but my assignment is asking me but I’m attempting to find the negative and positive zeros for this equation f(x) = 2x^5-3x^2+x-1

 

 

Answer 1

Descartes' Rule of Signs can be used to determine the possible number of positive real roots and the possible number of negative real roots of a polynomial.  To find the possible number of positive real roots, determine the number of times the sign changes between each polynomial term listed in order.  For the equation given, the signs of the terms are + - + -, so the sign changes 3 times.  Descartes's Rule says that there are either 3 or 1 positive real roots  (max is 3, and there can be lesser positive values differing by multiples of 2).  To determine the number of negative real roots, write the polynomial terms for f(-x), then determine the number of sign changes of it.  f(-x) = -2x^5 - 3x^2 - x -1, having 0 sign changes.  There are no negative real roots of this equation.

 

The rational roots theorem says that if rational zeros exist for a polynomial they must be +/- factors of the constant term divided by factors of the leading coefficient.  For the equation given, the possible rational roots are +/- 1 and +/- 1/2.  We know from Descartes' Rule of Signs that there are no negative real roots, so the only ones to be tested are 1/2 and 1.  Substituting 1 into the equation gives (f1) = -1, so 1 is not a root.  Substituting 1/2 into the equation gives f(1/2) = -7/16, so it is not a root either.  Thus the function given has no rational roots.  It must have at least 1 positive real root, and it must be irrational.

 

OK, so how does one find the real root(s)?  If you have a graphing calculator, you can plot the function, then find where it crosses the x axis.  If you do this, you will find that it only crosses the x-axis once, and it crosses at x approximately equal to 1.1314985.  The other four roots (since the order of the equation is 5, it has 5 roots) are complex. 

 

The easiest way to approximate all of the roots for this equation is to use the polynomial root finder app (PLYSMLT2) on a TI 84 graphing calculator.  The approximate roots given by this app are as follows:

 

-.7330534256 + .9658942147 i

-.7330534256 - .9658942147 i

1.131498516

.1673041674 + .5220640014 i

.1673041674 - .5220640014 i