1.4 - Factoring Polynomials - The Remainder Theorem MCB4U - Santowski

(A) Review to evaluate a polynomial for a given value of x, we simply substitute that given value of x into the polynomial. ex. Evaluate 4x 3 - 6x² + x - 3 for x = 2 we then have a special notations that we can write: for the polynomial; P(x) = 4x 3 - 6x² + x - 3 for substituting into the polynomial; P(2) = 4(2) 3 – 6(2)² + (2) - 3

(B) The Remainder Theorem Divide 3x 3 – 4x 2 - 2x - 5 by x + 1 Evaluate P(-1). What do you notice? if rewritten as 3x 3 – 4x 2 - 2x - 5 = (x + 1)(3x 5 - 7x + 5) - 10, notice P(-1) = -10 (Why?) Divide 6p p - 7 by 3p + 1 Evaluate P(-1/3). What do you notice? Rewrite the equation in “factored” form Divide 8p p + 5 by 2p - 5 Evaluate P(5/2). What do you notice? What must be true about (2p-5)? Divide x 2 - 5x + 4 by x - 4 Evaluate P(-4). What do you notice? What must be true about (x - 4)?

(B) The Remainder Theorem the remainder theorem states "when a polynomial, P(x), is divided by (ax - b), and the remainder contains no term in x, then the remainder is equal to P(b/a)

(C) Examples Find k so that when x 2 + 8x + k is divided by x - 2, the remainder is 3 Find the value of k so that when x 3 + 5x 2 + 6x + 11 is divided by x + k, the remainder is 3 When P(x) = a x 3 – x 2 - x + b is divided by x - 1, the remainder is 6. When P(x) is divided by x + 2, the remainder is 9. What are the values of a and b ?

(D) Internet Links Remainder Theorem and Factor Theorem from WTAMU The Remainder Theorem from The Math Page Remainder Theorem Lesson From Purple Math

(E) Homework Nelson text, page 50, Q2,3,4,9 (verify using RT), 10,11,14