Descartes' Rule of sign

Question

P(x) = 2x5 + 7x3 + 6x2 – 2
Possible Roots:      

 

Use Descartes' Rule of Signs to analyze the zeros of the following. List all possibilities.

 

please explain 

Answer 1

max 1 positive real root = about 0.46, slightly more than .46

max 2 negative real roots, 0 or 2 - real roots

2 or 4 imaginary roots

count the sign changes in terms = 1, so 1 + real root

replace x with -x and count the sign changes = 2 = maximum number of - real roots

imaginary roots come in conjugate pairs, so either 2 or 4 imaginary roots as real roots are 1 or 3

total roots are 5, the highest degree exponent

every odd degree polynomial has at least 1 real root = number of x intercepts