1. 5.6 The Remainder and Factor Theorems

2. [If you are dividing by (x - 6), the remainder will be the same as if you were evaluating the polynomial using synthetic substitution when x = 6]

3. Example 1: Synthetic Division Divide x 3 - 3x 2 - 7x + 6 by x + 2

4. Example 2: Synthetic Substitution If f(x) = 2x 4 – 5x 2 + 8x – 7, find f(6). Answer: The remainder is 2453. Thus, by using synthetic substitution, f(6) = 2453.

5. You try If f(x) = 2x 3 – 3x 2 + 7, find f(3). A.20 B.34 C.88 D.142

6. Example 4: Find Function Values The number of college students from the United States who study abroad can be modeled by the function S(x) = 0.02x 4 – 0.52x 3 + 4.03x 2 + 0.09x + 77.54, where x is the number of years since 1993 and S(x) is the number of students in thousands. How many U.S. college students will study abroad in 2011? Answer:In 2011, there will be about 451,760 U.S. college students studying abroad.

7. Example 5: The number of high school students in the United States who hosted foreign exchange students can be modeled by the function F(x) = 0.02x 4 – 0.05x 3 + 0.04x 2 – 0.02x, where x is the number of years since 1999 and F(x) is the number of students in thousands. How many U.S. students will host foreign exchange students in 2013? A.616,230 students B.638,680 students C.646,720 students D.659,910 students

8. Same format as synthetic substitution. Use FILLERS! used when divisor is in the form x - k. Create a polynomial out of coefficients that are left under the line. Factor further if possible. If the remainder is 0, then (x - k) is a factor of the polynomial k is called a zero because f(k) = 0

9. Example 6: Use the Factor Theorem Determine whether x – 3 is a factor of x 3 + 4x 2 – 15x – 18. Then find the remaining factors of the polynomial. Answer: So, x 3 + 4x 2 – 15x – 18 = (x – 3)(x + 6)(x + 1).

10. Check example 6

11. Example 7: Factor f(x) = 3x 3 + 13x 2 + 2x - 8 given that f(-4) = 0.

12. You Try Determine whether x + 2 is a factor of x 3 + 8x 2 + 17x + 10. If so, find the remaining factors of the polynomial. A.yes; (x + 5)(x + 1) B.yes; (x + 5) C.yes; (x + 2)(x + 3) D.x + 2 is not a factor.