The polynomial remainder theorem states,
If f(x) is a polynomial, then the remainder obtained by dividing f(x) by x-r
equals f(r)
In other words, x-r is a factor of f(x)-f(r)
To show this is true, we will first show that x-r is a factor of axn-arn,
where a is any real number, and n is a nonnegative integer. Then,
since the polynomial f(x)-f(r) is composed of a sum of numbers of the
form axn-arn, and x-r divides every term of that
sum, then it follows that x-r divides f(x)-f(r).
Before we begin, perhaps a definition of "remainder" is needed.
As you recall, in division of integers, if r is the remainder of the division p/d, then there is an integer, q, such that qd+r=p, and 0 <= r < d.
In division of polynomials the same idea holds, except the quotient and remainders are both polynomials.
In division of polynomials, if r is the remainder of the division p/d (both polynomials), then there is a polynomial, q, such that qd+r=p, and 0 <= O(r) < d.
Here O(r) means the "order" or "degree" of polynomial r.
Proof that (x-r) | (axn-arn) for all real a and
nonnegative integer n.
If (x-r) | (xn-rn) then (x-r) | (axn-arn),
so I'll just show the former.
If n=0, then x-r is a factor of xn-rn,
because x0-r0=0, and x-r is a factor of 0.
Now let's suppose x-r is a factor of xn-rn.
We will show x-r is a factor of xn+1-rn+1.
First, it is clear that (x-r) | (xn-rn)(x-r).
I will write a series of expressions that are all equivalent to (xn-rn)(x-r).
(x-r) | (xn-rn)(x-r)
(x-r) | xn+1-xnr-rnx+rn+1
by multiplying the terms together
(x-r) | xn+1-xnr+2rn+1-rnx-rn+1
by adding and subtracting 2arn+1.
(x-r) | xn+1-xnr+rnr+rnr-rnx-rn+1
by expressing 2arn+1 as arnr+arnr.
(x-r) | xn+1-r(xn-rn)-(x-r)(rn)-rn+1
by gathering terms, and factoring r and (x-r)
(x-r) | (xn+1-rn+1) - r(xn-rn)
- (x-r)(rn) by rearranging terms
(x-r) | r(xn-rn)
(x-r) | (x-r)(rn)
(x-r) | (xn+1-rn+1) (x-r)|a-b-c, (x-r)|b,
(x-r)|c, so (x-r)|a.
This proves that (x-r) | (xn-rn) for all nonnegative integer n.
Therefore (x-r) | (axn-arn) for all real a and nonnegative integer n.
Since the polynomial f(x)-f(r) is composed of a sum of numbers of the
form axn-arn, and x-r divides every term of that
sum, then it follows that x-r divides f(x)-f(r).
In other words,
If f(x) is a polynomial, then the remainder obtained by dividing f(x) by x-r
equals f(r)
