Question 38459
x^7-2x^6+3x^2-2x+5=0

The highest power of x is 7.  This is the degree or order of the equation.  It also tells you that there are a total of 7 roots, if you include real roots, complex roots, and repeated roots.  Complex roots will always be in conjugate pairs--i.e., two or four or six complex roots.  Therefore, these are the possibilites.


7 real and no complex roots
5 real and two complex roots (one conjugate pair).
3 real and four complex roots (two conjugate pairs).
1 real and six complex roots (three conjugate pairs).


8-4x^3+4x^6=0

The highest power of x is 6, so there will be a total of 6 roots.  These are the possibilities:

6 real roots and no complex roots
4 real roots and two complex roots (1 conjugate pair)
2 real and four complex roots (two conjugate pairs).
No real and six complex roots (three conjugate pairs).


R^2 at SCC