1. 3.2 The Remainder Theorem

2. Homework from last day P. 124 #1 – 5
And from Tuesday, p. P. 114 #1, 2, 3, 5, 6, 9, C4

3. The Remainder Theorem
Given P(x) = x3 - 4x2 + 5x + 1, determine the remainder
when P(x) is divided by x - 1. -1
The remainder is 3. -1 -2 3 1 -3 2 3
NOTE: P(1) gives
the same answer
as the remainder
using synthetic
division.
Using f(x) = x3 - 4x2 + 5x + 1, determine P(1):
P(1) = (1)3 - 4(1)2 + 5(1) + 1 =
= 3
Therefore P(1) is equal to the remainder. In other words,
when the polynomial x3 - 4x2 + 5x + 1 is divided by x - 1,
the remainder is P (1).

4. Remainder Theorem: When a polynomial P(x) is divided by x - a, the
remainder is P(a). [think x - a, then x = a]
Determine the remainder when x3 - 4x2 + 5x - 1 is divided by:
a) x - 2
b) x + 1
Calculate P(-1)
P(-1) = (-1)3 - 4(-1)2 + 5(-1) - 1 =
= -11
Calculate P(2)
P(2) = (2)3 - 4(2)2 + 5(2) - 1 =
= 1
The remainder is -11.
The remainder is 1.
Point (-1, -11) is on the graph of
of f(x) = x3 - 4x2 + 5x - 1
Point (2, 1) is on the graph of
of f(x) = x3 - 4x2 + 5x - 1

5. Applications When the remainder is 30. is divided by
Determine the value of k.

6. Problem Solving
When the polynomial 3x3 + ax2 + bx -9 is divided by x - 2 , the remainder is -5. When the polynomial is divided by x + 1, the remainder is -16. What are the values of a and b?

7. Assignment
Page 124
6a,7b, 8a,c, 9, 11, 14

8. Using Synthetic Division
1. (4x3 - 11x2 + 8x + 6) ÷ (x - 2)
P(x) = (x - 2)(4x2 - 3x + 2) + 10
- 8 6 -4 4 -3 2 10
2. (2x3 - 2x2 + 3x + 3) ÷ (x - 1)
P(x) = (x - 1)(2x2 + 3) + 6