Five-Minute Check (over Lesson 5–5) CCSS Then/Now New Vocabulary Key Concept: Remainder Theorem Example 1: Synthetic Substitution Example 2: Real-World Example: Find Function Values Key Concept: Factor Theorem Example 3: Use the Factor Theorem Over Lesson 5–6 Factor 8c3 – g3. If the polynomial is not factorable, write prime. A. (2c)(4c2 + cg + g2) B. (2c – g)(4c2 + 2cg + g2) C. (c – g)(2c + g + g2) D. prime Over Lesson 5–6 Factor 12az – 6bz – 6cz + 10ax – 5bx – 5cx. If the polynomial is not factorable, write prime. A. (2a – 3z)(5x – c) B. (2a – 6z)(2a + b + c) C. (5x + 6z)(2a – b – c) D. prime Over Lesson 5–6 Factor 8x3m2 – 8x3n2 + y3m2 – y3n2. If the polynomial is not factorable, write prime. A. (8x + y)(m + n)(m – n) B. (4x + y)(2x – y2)(m + n)(m – n) C. (2x + y)(4x2 – 2xy + y2)(m + n)(m – n) D. prime Over Lesson 5–6 Solve 16d4 – 48d2 + 32 = 0. A. B. C. D. Over Lesson 5–6 Solve k3 + 64 = 0. A. 16 B. 8 C. –2 D. –4 Over Lesson 5–6 The width of a box is 3 feet less than the length. The height is 4 feet less than the length. The volume of the box is 36 cubic feet. Find the length of the box. A. 2 ft B. 3 ft C. 4 ft D. 6 ft Content Standards A.APR.2 Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x). F.IF.7.c Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. Mathematical Practices 7 Look for and make use of structure. 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. • synthetic substitution • depressed polynomial Synthetic Substitution If f(x) = 2x4 – 5x2 + 8x – 7, find f(6). Method 1 Synthetic Substitution By the Remainder Theorem, f(6) should be the remainder when you divide the polynomial by x – 6. 2 2 0 –5 12 72 12 67 –7 Notice that there is no x3 term. A zero is 402 2460 placed in this position as a placeholder. 410 2453 8 Answer: The remainder is 2453. Thus, by using synthetic substitution, f(6) = 2453. Synthetic Substitution Method 2 Direct Substitution Replace x with 6. Original function Replace x with 6. Simplify. Answer: By using direct substitution, f(6) = 2453. If f(x) = 2x3 – 3x2 + 7, find f(3). A. 20 B. 34 C. 88 D. 142 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. 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? A. 616,230 students B. 638,680 students C. 646,720 students D. 659,910 students Use the Factor Theorem Determine whether x – 3 is a factor of x3 + 4x2 – 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. 1 4 –15 3 21 1 7 6 –18 18 0 Use the Factor Theorem Since the remainder is 0, (x – 3) is a factor of the polynomial. The polynomial x3 + 4x2 – 15x –18 can be factored as (x – 3)(x2 + 7x + 6). The polynomial x2 + 7x + 6 is the depressed polynomial. Check to see if this polynomial can be factored. x2 + 7x + 6 = (x + 6)(x + 1) Factor the trinomial. Answer: So, x3 + 4x2 – 15x – 18 = (x – 3)(x + 6)(x + 1). Use the Factor Theorem Check You can see that the graph of the related function f(x) = x3 + 4x2 – 15x – 18 crosses the x-axis at 3, –6, and –1. Thus, f(x) = (x – 3)[x – (–6)][x – (–1)]. Determine whether x + 2 is a factor of x3 + 8x2 + 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.