Question 1154820
{{{x^5 -3x^4 + x^3 -4x^2 + 5x - 1}}}

Here are the coefficients of our variable {{{x}}}:

{{{1}}}.....   {{{-3}}}.....     {{{1}}}.....     {{{ -4}}}.....    {{{ 5 }}}.....    {{{-1}}}

As can be seen, there are {{{5}}} changes.

This means that there are {{{5}}} or{{{ 3}}} or {{{1}}} {{{positive}}} real roots.

To find the number of {{{negative}}} real roots, substitute {{{x}}} with {{{-x}}} in the given polynomial:


 {{{x^5 -3x^4 + x^3 -4x^2 + 5x - 1}}} becomes {{{-x^5-3x^4-x^3-4x^2-5x-1}}}

The coefficients are {{{-1}}},{{{-3}}},{{{-1}}},{{{-4}}},{{{-5}}},{{{-1}}}.


As can be seen, there are{{{ 0}}} changes. This means that there are {{{0}}} {{{negative }}}real roots.


{{{ graph( 600, 600, -5, 5, -5, 5, x^5 -3x^4 + x^3 -4x^2 + 5x - 1) }}}


graph shows {{{3}}} {{{positive}}} real roots, means there will be one {{{pair}}} of {{{imaginary}}}  roots too