1.4 - Factoring Polynomials - The Remainder Theorem

MCB4U - Santowski

(A) Reviewto 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 - 3for substituting into the polynomial; P(2) = 4(2)3 – 6(2)² + (2) - 3

(B) The Remainder TheoremDivide 3x3 – 4x2 - 2x - 5 by x + 1Evaluate 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 + 1Evaluate P(-1/3). What do you notice? Rewrite the equation in “factored” form

Divide 8p2 - 11p + 5 by 2p - 5Evaluate P(5/2). What do you notice? What must be true about (2p-5)?

Divide x2 - 5x + 4 by x - 4Evaluate 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) ExamplesFind 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?

(D) Internet LinksRemainder Theorem and Factor Theorem from WTAMUThe Remainder Theorem from The Math PageRemainder Theorem Lesson From Purple Math

(E) Homework

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