1.4 - Factoring Polynomials - The Remainder Theorem

1. 1.4 - Factoring Polynomials - The Remainder Theorem MCB4U - Santowski

2. (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 4x3 - 6x² + x - 3 for x = 2 • we then have a special notations that we can write: • for the polynomial; P(x) = 4x3 - 6x² + x - 3 • for substituting into the polynomial; P(2) = 4(2)3 – 6(2)² + (2) - 3

3. (B) The Remainder Theorem • Divide 3x3 – 4x2 - 2x - 5 by x + 1 • Evaluate P(-1). What do you notice? • if rewritten as 3x3 – 4x2 - 2x - 5 = (x + 1)(3x5 - 7x + 5) - 10, notice P(-1) = -10 (Why?) • Divide 6p2 - 17p - 7 by 3p + 1 • Evaluate P(-1/3). What do you notice? Rewrite the equation in “factored” form • Divide 8p2 - 11p + 5 by 2p - 5 • Evaluate P(5/2). What do you notice? What must be true about (2p-5)? • Divide x2 - 5x + 4 by x - 4 • Evaluate P(-4). What do you notice? What must be true about (x - 4)?

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)

5. (C) Examples • Find k so that when x2 + 8x + k is divided by x - 2, the remainder is 3 • Find the value of k so that when x3 + 5x2 + 6x + 11 is divided by x + k, the remainder is 3 • When P(x) = ax3 – x2 - 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?

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

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