5.4 POLYNOMIALS 165For example, let f(x) = x^3 +x^2 + 7 and g(x) = x^2 + 3. Both polynomials are in
Z[x]. By doing "long division," we get
x^3 +x^2 + 7 = (x^2 + 3)(x+ 1) + (-3x+4),
so Q(x) = x+ I and R(x) = -3x + 4. The important thing is that the quotient Q(x) is
also in Z[x], i.e., also has integer coefficients. We may take the division algorithm for
granted, but it is a very important property of polynomials, as well as integers.
Example 5.4. 1 (AIME 1986) What is the largest integer n for which n^3 + 100 is di
visible by n + 1O?
Solution: Using the division algorithm, n 3 + 100 = (n + 1O)(n^2 - IOn + 100) -
900, so
n^3 + 100
= n^2 IOn + 100
900
n+1O n+1O
If n^3 + 100 is to be divisible by n + 10, then n��O must be an integer. The largest
positive n for which this is true is n = 890, of course. _
The Zeros of a Polynomial
It is always nice to solve a polynomial equation; undoubtedly you know the quadratic
formula, which states that if
cd-+bx+c = 0,
then
x=
(^2) a
While this formula is useful, it is far more important to remember how it was derived,
by using the complete the square tool. We will review this with a simple example.
x^2 + 6x - 5 = 0 {=:::} x^2 + 6x = 5 {=:::} x^2 + 6x + 9 = 14.
Thus (x+3f = 14, sox+3 = ±JI4, etc.
But often the exact zeros of a polynomial are difficult or impossible to determine,^9
and in fact, sometimes the exact zeros are not all that important, but rather indirect
information is what is needed. Thus it is important to understand as much as possible
about the relationship between the zeros of a polynomial and other properties. Here
are a few useful principles.
(^9) Formulas for the zeros of any cubic or quartic polynomial were discovered in the 16th century, and Abel
proved in the 19th century, that it is impossible, in general, to find an "elementary" formula for the zeros of all
5th-or higher-degree polynomials. See [41] for a very readable account of this.