1. Then/Now You used the Distributive Property and factoring to simplify algebraic expressions. Evaluate functions by using synthetic substitution. Determine whether a binomial is a factor of a polynomial by using synthetic substitution.

2. Vocabulary synthetic substitution – a convenient way to evaluate a function using synthetic division and the Remainder Theorem

3. Concept

4. Example 1 Method 1 Direct Substitution Replace x with 6. Answer: By using direct substitution, f(6) = 2453. Original function Replace x with 6. Simplify.

5. Example 1 Synthetic Substitution If f(x) = 2x 4 – 5x 2 + 8x – 7, find f(6). Method 2 Synthetic Substitution Answer: The remainder is 2453. Thus, by using synthetic substitution, f(6) = 2453. By the Remainder Theorem, f(6) should be the remainder when you divide the polynomial by x – 6. 21267410 2453 Notice that there is no x 3 term. A zero is placed in this position as a placeholder. 2 0–5 8 –7 1272402 2460

6. Example 1 A.20 B.34 C.88 D.142 If f(x) = 2x 3 – 3x 2 + 7, find f(3).

7. Example 2 Find Function Values COLLEGE 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:

8. Example 2 Find Function Values COLLEGE 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.

9. Example 2 A.616,230 students B.638,680 students C.646,720 students D.659,910 students HIGH SCHOOL 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?

10. Example 2 A.616,230 students B.638,680 students C.646,720 students D.659,910 students HIGH SCHOOL 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?

11. Concept

12. How do you solve Polynomial Equations that have variables raised to a power greater than two (e.g. cubic, quartic, quintic, etc.)? 1.First, break down equation into factors of powers less than 2 (e.g. sum or difference of cubes, quadratic form 2.Set each factor equal to zero and solve Today we are going to talk about one way to do the first step.

13. Example 3 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. The binomial x – 3 is a factor of the polynomial if 3 is a zero of the related polynomial function. Use the factor theorem and synthetic division. 17601760 14–15–18 32118

14. Example 3 Use the Factor Theorem Since the remainder is 0, (x – 3) is a factor of the polynomial. The polynomial x 3 + 4x 2 – 15x –18 can be factored as (x – 3)(x 2 + 7x + 6). The polynomial x 2 + 7x + 6 is the depressed polynomial. Check to see if this polynomial can be factored. x 2 + 7x + 6 = (x + 6)(x + 1)Factor the trinomial. Answer:

15. Example 3 Use the Factor Theorem Since the remainder is 0, (x – 3) is a factor of the polynomial. The polynomial x 3 + 4x 2 – 15x –18 can be factored as (x – 3)(x 2 + 7x + 6). The polynomial x 2 + 7x + 6 is the depressed polynomial. Check to see if this polynomial can be factored. x 2 + 7x + 6 = (x + 6)(x + 1)Factor the trinomial. Answer: So, x 3 + 4x 2 – 15x – 18 = (x – 3)(x + 6)(x + 1).

16. Example 3 Use the Factor Theorem CheckYou can see that the graph of the related function f(x) = x 3 + 4x 2 – 15x – 18 crosses the x-axis at 3, –6, and –1. Thus, f(x) = (x – 3)[x – (–6)][x – (–1)].

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

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