If there is one variable. Then put each variable equal to zero and then solve for the other variable.
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A quadratic polynomial must have zeros, though they may be complex numbers.A quadratic polynomial with no real zeros is one whose discriminant b2-4ac is negative. Such a polynomial has no special name.
If the cubic polynomial you are given does not have an obvious factorization, then you must use synthetic division. I'm sure wikipedia can tell you all about that.
The values of the variables which make the polynomial equal to zero
In the real domain, yes. In the complex domain, no.
Either graph the polynomial on graph paper manually or on a graphing calculator. If it is a "y=" polynomial, then the zeroes are the points or point where the polynomial touches the x-axis. If it is an "x=" polynomial, then the zeroes are the points or point where the polynomial touches the y-axis. If it touches neither, then it has no zeroes.