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Chapter 8 POLYNOMIAL EQUATIONS
An nth degree polynomial with complex coefficients is of the form a₀xⁿ + a₁xⁿ^-1 + ... + a_n_-1x + a_n where the a_i are complex numbers and

(Of course, the coefficients a₀, ..., a_n may be real numbers, since a real number is a special case of a complex number.)

Thus a first degree polynomial is of the form ax + b and a second degree polynomial of the form ax² + bx + c, with in both cases. A non-zero constant a is a polynomial of degree zero; the constant zero is also a polynomial, but it is not assigned a degree.

The polynomial equation y = x² - 6x + 1 defines y to be a function of x on the domain of all complex numbers; that is, it provides a rule for assigning a unique complex number y to each complex number x. The table

x	4	3	2	1	0	i	2i
y = x² - 6x + 1	-7	-8	-7	-4	1	-6i	-3 - 12i

shows that this functional rule assigns -4 to 1, 1 to 0 , -6i to i, etc. A rule that makes y a function of x assigns precisely one value y to a fixed x; however, the same number y may be assigned to more than one x, as is seen here with -7 assigned to 2 and to 4.

It is sometimes convenient to represent the rule that defines y to be a function of x by the symbol f(x). This notation enables one to express in a simple way the number assigned to a given x by the function. For example, f(1), f(2), and f(3) stand for the numbers assigned to 1, 2, and 3, respectively. If f(x) = x² - 6x + 1, then f(1) = -4, f(2) = -7, f(3) = -8,

f(a + b) = (a + b)² - 6(a + b) + 1, and f(x + 1) = (x + 1)² - 6(x + 1) + 1.
Notice that f(a + b) is not necessarily the same as f(a) + f(b), since f(a + b) is the result of replacing x in x² - 6x + 1 by a + b and is not f times a + b.

If several functions are involved in a given discussion, one may use g(x), F(x), p(x), q(x), and so on, as alternates for f(x).

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Tuesday, June 2, 1998