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 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
(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)?
(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 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 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