What is the greatest number that is a factor each of two or more numbers?
it depends on the power of the leading coefficient, and that is
not always a great indication because polynomials can have non real
numbers.
A factor of a polynomial is where the function crosses the x
axis.
If the trinomial will not factor into real numbers, then there
are not any real zeros but there are still factors.
Think of this one x^2+6x+14. this will not factor into real
numbers, but complex solutions. But these complex solutions are
factors, so the rule still holds.
If the trinomial is a cubic, or at a odd power, then its a odd
function, and can have one real solution.
If the trinomial is squared, or any even power, its a even
function and can have two real solutions.
With the graph you can determine it this way:
if p(x) is a polynomial function of degree n, then the graph has
at most n-1 turning points.
If the graph of a function P has n-1 turning points, then the
degree of p(x) is at least n.