1.
P(x) = sum a_i x^i for i = 0 .. N
The multipliers, a_i, are known as "coefficients" and N, the highest power of x with a non-zero coefficient, is known as the "degree" of the polynomial. If N=0 then P(x) is constant, if N=1, P(x) is linear in x. N=2 gives a "quadratic" and N=3, a "cubic".
2.