zero (of a function)

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: −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 abi).

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