Question 288534
<pre><font size = 4 color = "indigo"><b>
{{{f(x)=x^5-1.7x^4-17.65x^3+3x^2+45x-15.411}}}

The 1st term is {{{""+x^5}}}, That has a positive sign.

The 2nd term is {{{-1.7x^4}}}.  That has a negative sign.  Therefore
going from the positive 1st term to the negative 2nd term is a sign 
change.  So far there is 1 sign change.

The 3rd term is {{{-17.65x^3}}}.  That has a negative sign.  Therefore
going from the negative 2nd term to the negative 3rd term is NOT a sign 
change.  So far there is still just 1 sign change.

The 4th term is {{{""+3x^2}}}.  That has a positive sign.  Therefore
going from the negative 3rd term to the positive 4th term is a sign 
change.  So far there are now 2 sign changes.

The 5th term is {{{""+45x}}}.  That has a positive sign.  Therefore
going from the positive 4th term to the positive 5th term is NOT a sign 
change.  So far there are still just 2 sign changes.

The 6th term is {{{-15.411}}}.  That has a negative sign.  Therefore
going from the positive 5th term to the negative 6th term is a sign 
change.  So far there are now 3 sign changes.

We have reached the end of the polynomial.  It has 3 sign changes.

So there could be 3 positive zeros.  There could also be 1 positive
zero, because the number of positive zeros could either be the number
of sign changes, or a multiple of 2 fewer positive zeros. 

So there are either 3 or 1 positive zero.

Next substitute {{{-x}}} for {{{x}}} and simplify:

{{{f(-x)=(-x)^5-1.7(-x)^4-17.65(-x)^3+3x^2+45x-15.411}}}

{{{f(-x)=-x^5-1.7x^4+17.65x^3+3x^2-45x-15.411}}}

The 1st term is {{{-x^5}}}, That has a negative sign.

The 2nd term is {{{-1.7x^4}}}.  That has a negative sign.  Therefore
going from the negative 1st term to the negative 2nd term is NOT a sign 
change.  So far there are no sign changes.

The 3rd term is {{{""+17.65x^3}}}.  That has a positive sign.  Therefore
going from the negative 2nd term to the positive 3rd term is a sign 
change.  So far there is now just 1 sign change.

The 4th term is {{{""+3x^2}}}.  That has a positive sign.  Therefore
going from the positive 3rd term to the positive 4th term is NOT a sign 
change.  So far there is still just 1 sign change.

The 5th term is {{{-45x}}}.  That has a negative sign.  Therefore
going from the positive 4th term to the negative 5th term is a sign 
change.  So far there are now 2 sign changes.

The 6th term is {{{-15.411}}}.  That has a negative sign.  Therefore
going from the negative 5th term to the negative 6th term is NOT a sign 
change.  So far there are still 2 sign changes.

So there could be 2 negative zeros.  There could also be 0 negative
zeros, because the number of negative zeros could either be the number
of sign changes, or a multiple of 2 fewer negative zeros. 

So there are either 2 or 0 negative zeros.

Edwin</pre>