Complexe nulpunten

Complex zeros are values that make a polynomial equal to zero and could appear during factoring when the solutions include imaginary components.

Because the complex number system extends the real numbers, any polynomial with real or complex coefficients can be fully factored using complex zeros.

This relationship is formalized in the Complete Factorization Theorem, which states that a polynomial of degree n, with coefficients labeled by subscripts to indicate each term’s position, can be written as a product of linear factors, with each factor corresponding to a complex zero.

Some of these zeros may occur more than once—this property is known as multiplicity.

For instance, a zero with multiplicity three has its factor raised to the third power in the polynomial.

When multiplicities are counted, the Zeros Theorem states that a polynomial of degree n has exactly n zeros, which may be real, complex, or repeated.

A practical use of complex zeros is in notch filters, which block specific frequencies. Medical instruments often pick up noise from power lines at 50 or 60 Hz. Placing complex zeros at those frequencies removes the noise while preserving the rest of the signal.