
Glossary: Prime Pages: Top 5000: 
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: 3pi, 2pi, pi, pi, 2pi, 3pi,. . .
The multiplicity of a zero of a polynomial is how often it occurs. For example, the zeros of (x3)^{2}(x4)^{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 abi).
Chris K. Caldwell © 19992018 (all rights reserved)
