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2 1. Fundamentals

constant polynomial has degree 0, but, by convention, the zero polyno-

mial (all coefficients vanishing) has degree -oo. Special names are given to

polynomials of low degree:

degree of polynomial type of polynomial

1 linear

2 quadratic

3 cubic

4 quartic

5 quintic

We can evaluate a polynomial by replacing its variable by any number
and carrying out the computation. The value of a polynomial p(t) at t = r
is denoted by p(r). For example, if p(t) = 3t3 - 2t2 - t + 4, its value when
t = 2 is p(2) = 3.23 - 2.22 - 2 + 4 = 24 - 8 - 2 + 4 = 18. Since polynomials
are a simple type of function easy to evaluate, they are very useful in
approximating other more complex functions.
A zero of a polynomial p(t) is any number r for which p(r) takes the
value 0. When p(r) = 0, we say that r is a root or a solution of the
equation p(t) = 0. There are many situations in which we need to have
information about the zeros of a polynomial, and considerable amount of
attention is devoted to methods of solving equations p(t) = 0 either exactly
or approximately. In particular, knowing the zeros of polynomials is often
helpful in graphing a wide variety of functions and obtaining inequalities.
In operating with polynomials, we treat the variables as though they
were numbers. Let

p(t) = a0 + alt + a2t2 +... + a,t”

q(t) = b. + bit + b2t2 +... + b,tm

be any two polynomials.

Sum: (p + q)(t) = (a0 + bo) + (UI + bl)t + (a2 + b2)t2 +. ...

Difference: (p - q)(t) = (a0 - bo) + (al - b,)t + a... a

Product of a constant and a polynomial: (cp)(t) = cao + calt + ca2t2 +....

Product of two polynomials: (pq)(t) = aobo + (aobl + albo)t +

(aoba+albl +a2bo)t2 +***+(aob, +alb,-1 +...+aib,-i+...+a,bo)t’+

... + (u,b,)tm+?
Composition of two polynomials: (p o q)(t) = p(q(t)). This definition in-
structs us to replace each occurrence oft in the expression for p(t) by q(t).