A zero or root (archaic) of a function is a
value which makes it zero. For example, the zeros of
x2-1 are x=1 and x=-1. The
zeros of z2+1 are z=i and
z=-i. Sometimes we restrict our domain, so limiting
what type of zeros we will accept. For example,
z2+1 has no real zeros (because its two
zeros are not real numbers). x2-2 has no
rational zeros (its two zeros are irrational
numbers). The sine function has no algebraic zeros
except 0, but has
infinitely many transcendental zeros: -3pi, -2pi, -pi, pi, 2pi, 3pi,. . .
The multiplicity of a zero of a polynomial is how often it occurs. For example, the zeros of (x-3)2(x-4)5 are 3 with multiplicity 2 and 4 with multiplicity 5. So this polynomial has two distinct zeros, but seven zeros (total) counting multiplicities.
The fundamental theorem of algebra states that a polynomial (with real or complex coefficients) of degree n has n zeros in the complex numbers (counting multiplicities). It then follows that a polynomial with real coefficients and degree n has at most n real zeros. Finally, the complex zeros of a polynomial with real coefficients come in conjugate pairs (that is, if a+bi is a zero, then so is a-bi).