4.4 Theorems about Zeros of Polynomial Functions

  • Find a polynomial with specified zeros.

  • For a polynomial function with integer coefficients, find the rational zeros and the other zeros, if possible.

  • Use Descartes’ rule of signs to find information about the number of real zeros of a polynomial function with real coefficients.

We will now allow the coefficients of a polynomial to be complex numbers. In certain cases, we will restrict the coefficients to be real numbers, rational numbers, or integers, as shown in the following examples.

Polynomial Type of Coefficient
5x33x2+(2+4i)x+i Complex
5x33x2+2xπ Real
5x33x2+23x74 Rational
5x33x2+8x11 Integer

The Fundamental Theorem of Algebra

A linear, or first-degree, ...

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