1. 6.5 The Remainder and Factor Theorems

2. Dividing Polynomials Using polynomial long division
Divide y4 + 2y2 – y + 5 by y2 – y + 1
Divide x3 – x2 – 2x + 8 by x – 1
Divide x3 – x2 – 2x + 8 by x + 2
Divide 2x2 + 13x – 7 by x + 6

3. Examples: Using Long Division (x3 – 2x2 – 8x + 5) ÷ (x – 1)

4. Investigation
f(x)=3x3 – 2x2 + 2x – 5 1) Divide the function by x – 2. What is the quotient? What is the remainder? 2) Use synthetic substitution to evaluate f(2). How is f(2) related to the remainder? What do you notice about the other constants in the last row of synthetic substitution?

5. Remainder Theorem
If a polynomial f(x) is divided by x – k, then the remainder is r = f(k)

6. Synthetic Division
Synthetic substitution gives you the coefficients of the quotient
Synthetic Substitution often called synthetic division

7. Examples Using Synthetic Division (x3 – 3x2 – 7x + 6) ÷ (x + 2)

8. Examples Using Synthetic Division
(3x4 – 17x3 + 13x2 – 24x + 16) ÷ (x + 4)
(4x4 – 5x3 + 2x2 – x + 5) ÷ (x – 2)

9. Factor Theorem
A polynomial f(x) has a factor x – k if and only if f(k) = 0
Examples:
Factor f(x) = 3x3 + 13x2 + 2x – 8 given that f(-4) = 0
Factor f(x) = 3x3 + 14x2 – 28x – 24 given that f(-6)=0

10. Finding Zeros of a Polynomial Function
One zero of f(x) = x3 + 6x2 + 3x – is x = -5. Find the other zeros of the function.
One zero of f(x) = 2x3 – 9x2 – 32x – 21 is x = 7. Find the other zeros of the function.

11. Examples
A company that manufactures CD – ROM drives would like to increase its production. The demand function for the drives is p = 75 – 3x2, where p is the price the company charges per unit when the company produces x million units. It costs he company $25 to produce each drive.

12. Examples
a) Write an equation giving the company’s profit as a function of the number of CD – ROM drives it manufactures.

13. Examples
b) The company currently manufactures 2 million CD – ROM drives and makes a profit of $76, 000, At what other level of production would the company also make $76, 000, 000?