Question 1089558
The rule states that if the terms of a single-variable polynomial with real coefficients are ordered by descending variable exponent, then the number of positive roots of the polynomial is either equal to the number of sign differences between consecutive nonzero coefficients, or is less than it by an even number.

So, coefficients are {{{2}}},{{{-5}}},{{{1}}},{{{4}}}.

As can be seen there are {{{2}}} changes; + (in front of 2)  changes to -(in front of 5), then -(in front of 5) to +(in front of 1), and + (in front of 1) does not change because next sign is also + (in front of 4)

This means that there are {{{2}}} or {{{0}}} positive real roots.


To find number of negative real roots substitute {{{x}}} with {{{-x}}} in the given polynomial: {{{F(x)=2x^3-5x^2+x+4}}} becomes {{{F(-x)=-x^3-5x^2-x+4}}}.

Coefficients are {{{-2}}},{{{-5}}},{{{-1}}},{{{4}}}.

As can be seen there is {{{1}}} change; 
all signs are -, only - (in front of 1) changes to +(in front of 4)

This means that there is {{{1}}} negative real root.

Answer:

*{{{2}}} or {{{0}}} positive real roots;{{{1}}} negative real root