1. Advanced Algebra Notes Section 5.5: Apply the Remainder and Factor Theorems Common Core Standard: A-APR #2 Know and apply the Remainder Theorem

2. Advanced Algebra Notes Section 5.5: Apply the Remainder and Factor Theorems Remember when you used to use long division when solving a basic general math problem like:

3. Definitions from the problem above: 1. Dividend: 2. Divisor: 3. Quotient: 4. Remainder: Number inside the division box (436) Number outside the division box (5) Number above the division box ( ) Number that gets written as the numerator of the fraction (1) We are going to do something very similar to that but with polynomials, and it is called _________________. long division

4. Examples:Divide these polynomials using long division. 1.

5. 2)

6. We can use another method to divide polynomials as long as the divisor is in the form x – k. This method is called _________________, which is exactly the same method as synthetic substitution that we did in section 5.2. synthetic division Remainder Theorem: If a polynomial f(x) is divided by x – k, then the remainder is r = f(k). Example:Divide using synthetic division. 3)Divide f(x) = 2x 3 + 9x 2 + 14x + 5 by x – 3 2 9 14 5 2 6 15 45 59 177 182

7. 4) Divide f(x) = -4x 3 + 5x 2 + 8 by x + 3

8. Suppose the remainder is 0 when a polynomial f(x) is divided by x – k: so x – k is a __________of the dividend f(x). Factor Theorem:A polynomial f(x) has a factor x – k if and only if f(k) = 0. The factor theorem can be used to solve a variety of problems. ProblemExample Given one factor of a polynomial, find the other factors. Example 5 below Given one zero of a polynomial function, find the other zeros. Example 6 Examples:Factor completely. 5) f(x) = 2x 3 – 11x 2 + 3x + 36 given that x – 3 is a factor factor -15 -12 -36 0 Fix this

9. 6) One zero of f(x) = x 3 + x 2 – 16x – 16 is 4. What is another zero of f(x)?

10. A company’s profit C (in thousands of dollars) can be modeled by C = -5x 3 + 6x 2 + 15x, where x is the number of items produced in thousands. The profit is $14,000 for producing 2000 items. What other number of items would produce about the same profit?