# zero (of a function)

A **zero** or **root** (archaic) of a function is a
value which makes it zero. For example, the zeros of
*x*^{2}−1 are *x*=1 and *x*=−1. The
zeros of *z*^{2}+1 are *z*=i and
*z*=−i. Sometimes we restrict our domain, so limiting
what type of zeros we will accept. For example,
*z*^{2}+1 has no real zeros (because its two
zeros are not real numbers). *x*^{2}−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: −3π, −2π, −π, π, 2π, 3π,. . .

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*).