Dr. Atul Nischal
The Remainder Theorem
Suppose p(x) is a polynomial of degree at least 1 and c is a real number. When p(x) is divided by x – c the remainder is p(c).
Proof
Let’s first recall the division algorithm for polynomials.
Suppose d(x) and p(x) are nonzero polynomials where the degree of p is greater than or equal to the degree of d. There exist two unique polynomials, q(x) and r(x), such that
p(x) = d(x) q(x) + r(x),
where either r(x) = 0 or the degree of r is strictly less than the degree of d.
When a polynomial is divided by x – c, the remainder is either 0 or has degree less than the degree of x – c.
Since x – c is degree 1, the degree of the remainder must be 0, which means the remainder is a constant.
Hence, in either case, p(x) = (x – c) q(x) + r, where r, the remainder, is a real number, possibly 0.
It follows that
p(c) = (c – c) q(c) + r
= 0 ´ q(c) + r
= r
So, we get r = p(c) as required.