Quick Polynomial Question?

Question

Considering: f(x) = 2x^5 - 3x^4 - 2x^3 +7x^2 - 2x - 6
Solve: f(x) is greater than or equal to 0.

Answer 1

f(x) = 2x^5 - 3x^4 - 2x^3 +7x^2 - 2x - 6
If f(x) =0,
f(0) = 2*0^5 - 3*0^4 - 2*0^3 + 7*0^2 - 2*0 - 6
=0 - 0 - 0 + 0 - 0 -6
= - 6

If f(x)>0,
ie, f(x) may be, 1,2,3,4,...................
So, like f(0), you may find the others

For example,
f(1) = 2*1^5 - 3*1^4 - 2*1^3 + 7*1^2 - 2*1 - 6
= 2^5 - 3^4 - 2^3 + 7^2 - 2 - 6
=32 - 81 - 8 + 49 -8
= 81-81-16
=0 -16
= - 16

Answer 2

first you need to check if the polynomial factorizes over integers Z. (i.e. try to find if -6,-3,-2,-1,1,2,3 are roots).
Now the derivative of f is:
f'(x) = 2((x+1)(5x^3 - 11x^2 + 8x - 1)) , which you can see further that has 4 real roots. This means that f suffers 4 changes of monotony which also implies that f has 5 REAL roots.
Call it a,b,c,d,e in increasing order.
Then the intervals where f is greater or equal than 0 will be:
[a,b], [c,d], [e, +infinity]. Why? Now you find approximations of the roots by using, say, Newton approximations.
I hope computations are right.

P.S. There might be other methods, like Sturm methods, but I don't have them handy now.