Question

P(x)= x^5-4x^4+3x^3+2x-6 Use Descartes rule of signs to analyze the zeros

Answer 1

$\bf \begin{array}{cccccccccccllll} P(x)= x^5&-&4x^4&+&3x^3&+&2x&-&6\\ &yes&&yes&&\textit{no}&&yes \\\\\\ P(-x)= -x^5&-&4x^4&-&3x^3&-&2x&-&6\\ &\textit{no}&&\textit{no}&&\textit{no}&&\textit{no} \end{array}$

notice where how many "yes" sign change happen in the positive "x"
and the on the negative "-x"

for the P(x) is 3 changes, thus
so is either 3 real positive ones, or 3-2 = 1 real positive one

P(-x) gives no sign changes, so there are 0 negative ones

the polynomial has a degree of 5, thus 5 roots
if it has 3 real positive ones, then it has 2 complex ones
the complex ones pick up the slack per se

if it has 1 real positive one, then it has 4 complex ones

bear in mind complex ones always come in pairs