166 CHAPTER 5 ALGEBRA
The Remainder TheoremIf the polynomial P(x) is divided by x - a the remainder will be P(a).
For example, divide x^3 - 2.x + 3 by x + 2 and get (after some work)
x^3 - 2.x+3 2 1
--....,.----=.r -2.x+2---;
x+2 x+2
i.e., the quotient is y. - 2x + 2 and the remainder is -1. And indeed,
(_2)^3 - 2( -2) +3 =- 1.
To see why the Remainder Theorem is true in general, divide the polynomial P(x) by
x - a, getting quotient Q(x) with remainder r. Using the division algorithm, we write
P(x) = Q(x) (x - a) + r.
The above equation is an identity; i.e., it is true for all values of x. Therefore we are
free to substitute in the most convenient value of x, namely x = a. This yields P(a) = r,
as desired. Please make a note of this substitute convenient values tool. It has many
applications!The Factor TheoremIf a is a zero of a polynomial P(x), then x - a must be a fa ctor; i.e.,
P(x) is a product of x - a and another polynomial.
This follows immediately from the Remainder Theorem.The Fundamental Theorem of AlgebraThe Factor Theorem above tells us that x - a is a factor of the polynomial P(x) if a
is a zero. But how do we know if a polynomial even has a zero? The Fundamental
Theorem of Algebra guarantees this:Every polynomial in C[x] has at least one complex zero.
This theorem is quite deep and surprisingly hard to prove. Its proof is beyond thescope of this book.^10
A corollary of the Fundamental Theorem (use the result of Problem 5 .4.6 below)is that any nth-degree polynomial has exactly n complex zeros, although some of the
zeros may not be distinct. Thus we have the following factored form for any polyno
mial :anx" + an_ 1 x"-^1 + ... +ao = an(x - rl)(x-r 2 )'" (x - rn), (2)
where the ri are the zeros, possibly not all distinct.
If zeros are not distinct, we say that they have multiplicity greater than 1. For
example, the 8th-degree polynomial(x - 1 )(x - 2i)(x + 2i)(x - 7)^3 (x+ 6 )^2
IOFor an elementary but difficult proof, see [9]. For a much simpler but less elementary argument, see [29].