#### Answer

The possible number of negative roots of a polynomial equation can be determined by calculating the number of sign changes using Descartes’s Rule of Signs.

#### Work Step by Step

To find possibilities for negative real zeros, count the number of sign changes in the equation for f (-x). We obtain this equation by replacing x with -x in the given function.
For example, consider the polynomial function\[f\left( x \right)={{x}^{3}}+2{{x}^{2}}+5x+4\]…… (1)
Replace x with –x in equation (1).
\[\begin{align}
& f\left( -x \right)={{\left( -x \right)}^{3}}+2{{\left( -x \right)}^{2}}+5\left( -x \right)+4 \\
& f\left( -x \right)=-{{x}^{3}}+2{{x}^{2}}-5x+4 \\
\end{align}\]
The equation with \[f\left( -x \right)\]is
\[f\left( -x \right)=-{{x}^{3}}+2{{x}^{2}}-5x+4\] ….. (2)
Now, we see that there are three variations in sign.
The number of negative real zeros of f (x) is either equal to the number of sign changes, 3, or is less than this number by an even integer. This means that either there are 3 negative real zeros or there is 3 - 2 = 1 negative real zero.