We next use this theorem to obtain information concerning the case in which r is not a root of p(x).

THE REMAINDER THEOREM: Let p(x) be a polynomial. Then for every complex number r there is a polynomial q(x) such that

(2)                     p(x) = (x - r)q(x) + p(r)

Then f(r) = p(r) - p(r) = 0. Hence r is a root of f(x) and, by the Factor Theorem, above, x - r is a factor of f(x), and so there is a polynomial q(x) such that
 

Now p(x) - p(r) = (x - r)q(x), since both sides are equal to f(x); equation (2) is then obtained by transposing p(r).

The polynomial p(r) is the remainder in the division of p(x) by x - r. In specific cases, the quotient polynomial q(x) of (2), above, may be found by long division or by a more compact form of division called synthetic division. We first illustrate these techniques on the example in which
 

Dividing p(x) by x - 2, we have
 

That is, p(x) = (x - 2)q(x) + p(2), with q(x) = x² - 5x - 6 and p(2) = -3.