The Remainder Theorem Consider = (x + f(x) r)q(x) R − Note that if we let the expression becomes x = r, = (r + f(r) r) q(r) R − Simplifying gives: = f(r) R This leads us to the Remainder Theorem which states: R If a polynomial is divided by (x and a remainder is obtained, then = f(x) r) f(r) R. − Example 3 Use the remainder theorem to find the remainder for Example 1 above, which was divide = 3x2 + 5x 8 by (x 2). f(x) − − Example 4 By using the remainder theorem, determine the remainder when 3x3 20x + 5 x2 − − is divided by (x + 4).
The Remainder Theorem Consider f(x) = (x − r)q(x) + R Note that if we let x = r, the expression becomes f(r) = (r − r) q(r) + R Simplifying gives: f(r) = R This leads us to the Remainder Theorem which states: If a polynomial f(x) is divided by (x − r) and a remainder R is obtained, then f(r) = R. Example 3 Use the remainder theorem to find the remainder for Example 1 above, which was divide f(x) = 3x2 + 5x − 8 by (x − 2). Example 4 By using the remainder theorem, determine the remainder when 3x3 − x2 − 20x + 5 is divided by (x + 4).
rem_fac_theorems.docx