Algebra and Trigonometry, 5th Edition

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.

JUST IN TIME 1

Polynomial	Type of Coefficient
$5x^3-3x^2+(2+4i)x+i$	Complex
$5x^3-3x^2+\sqrt{2}x-\pi$	Real
$5x^3-3x^2+\frac{2}{3}x-\frac{7}{4}$	Rational
$5x^3-3x^2+8x-11$	Integer

The Fundamental Theorem of Algebra

A linear, or first-degree, ...