Question 127623
{{{4x^3-9x^2+2x+3=0}}}

The signs are + - + +.  So there are 2 sign changes, 1st term to 2nd term, and 2nd term to 3rd term.


Two sign changes means that there are at most 2 positive roots.  There could also be 0 positive roots.


Evaluate {{{f(-x)}}}
{{{4(-x)^3-9(-x)^2+2(-x)+3=0}}}
{{{-4x^3-9x^2-2x+3=0}}}


The signs are - - - +.  So there is 1 sign change from the 3rd to the 4th term.


One sign change on {{{f(-x)}}} means that there is exactly 1 negative real root.


So overall, this equation has either 2 positive real roots and 1 negative real root, or a conjugate pair of complex roots and 1 negative real root.


A graph of the function demonstrates that the first possibility is the actual case.


{{{graph(600,600,-5,5,-5,5,4x^3-9x^2+2x+3)}}}