NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 19 M1 ALGEBRA II Lesson 19: The Remainder Theorem Student Outcomes . Students know and apply the remainder theorem and understand the role zeros play in the theorem. Lesson Notes In this lesson, students are primarily working on exercises that lead them to the concept of the remainder theorem, the connection between factors and zeros of a polynomial, and how this relates to the graph of a polynomial function. Students should understand that for a polynomial function 푃 and a number 푎, the remainder on division by 푥 − 푎 is the value 푃(푎) and extend this to the idea that 푃(푎) = 0 if and only if (푥 − 푎) is a factor of the polynomial (A-APR.B.2). There should be plenty of discussion after each exercise. Classwork Exercises 1–3 (5 minutes) Assign different groups of students one of the three problems from this exercise. Scaffolding: Have them complete their assigned problem, and then have a student from each If students are struggling, replace the group put their solution on the board. Having the solutions readily available allows polynomials in Exercises 2 and 3 with students to start looking for a pattern without making the lesson too tedious. easier polynomial functions. Examples: Exercises 1–3 2 푔(푥) = 푥 − 7푥 − 11 1. Consider the polynomial function 풇(풙) = ퟑ풙ퟐ + ퟖ풙 − ퟒ. a. Divide by 푥 + 1. b. Find 푔(−1). a. Divide 풇 by 풙 − ퟐ. b. Find 풇(ퟐ). 3 풇(풙) ퟑ풙ퟐ + ퟖ풙 − ퟒ 풇(ퟐ) = ퟐퟒ (푥 − 8) − 푔(−1) = −3 = 푥 + 1 풙 − ퟐ 풙 − ퟐ ퟐퟒ = (ퟑ풙 + ퟏퟒ) + 풙 − ퟐ 2 ℎ(푥) = 2푥 + 9 a. Divide by 푥 − 3. b. Find ℎ(3). 2. Consider the polynomial function 품(풙) = 풙ퟑ − ퟑ풙ퟐ + ퟔ풙 + ퟖ. 27 (2푥 + 6) + ℎ(3) = 27 a. Divide 품 by 풙 + ퟏ. b. Find 품(−ퟏ). 2푥2 + 9 품(풙) 풙ퟑ − ퟑ풙ퟐ + ퟔ풙 + ퟖ 품(−ퟏ) = −ퟐ = 풙 + ퟏ 풙 + ퟏ ퟐ = (풙ퟐ − ퟒ풙 + ퟏퟎ) − 풙 + ퟏ Lesson 19: The Remainder Theorem 206 This work is licensed under a This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from ALG II-M1-TE-1.3.0-07.2015 Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 19 M1 ALGEBRA II 3. Consider the polynomial function 풉(풙) = 풙ퟑ + ퟐ풙 − ퟑ. a. Divide 풉 by 풙 − ퟑ. b. Find 풉(ퟑ). 풉(풙) 풙ퟑ + ퟐ풙 − ퟑ 풉(ퟑ) = ퟑퟎ = 풙 − ퟑ 풙 − ퟑ ퟑퟎ = (풙ퟐ + ퟑ풙 + ퟏퟏ) + 풙 − ퟑ Discussion (7 minutes) . What is 푓(2)? What is 푔(−1)? What is ℎ(3)? 푓(2) = 24; 푔(−1) = −2; ℎ(3) = 30 . Looking at the results of the quotient, what pattern do we see? MP.8 The remainder is the value of the function. Stating this in more general terms, what do we suspect about the connection between the remainder from dividing a polynomial 푃 by 푥 − 푎 and the value of 푃(푎)? The remainder found after dividing 푃 by 푥 − 푎 will be the same value as 푃(푎). 13 . Why would this be? Think about the quotient . We could write this as 13 = 4 ∙ 3 + 1, where 4 is the 3 quotient and 1 is the remainder. Apply this same principle to Exercise 1. Write the following on the board, and talk through it: 푓(푥) 3푥2 + 8푥 − 4 24 = = (3푥 + 14) + 푥 − 2 푥 − 2 푥 − 2 . How can we rewrite 푓 using the equation above? Multiply both sides of the equation by 푥 − 2 to get 푓(푥) = (3푥 + 14)(푥 − 2) + 24. In general we can say that if you divide polynomial 푃 by 푥 − 푎, then the remainder must be a number; in fact, there is a (possibly non-zero degree) polynomial function 푞 such that the equation, 푃(푥) = 풒(풙) ∙ (푥 − 푎) + 풓 ↑ ↑ quotient remainder is true for all 푥. What is 푃(푎)? 푃(푎) = 푞(푎)(푎 − 푎) + 푟 = 푞(푎) ∙ 0 + 푟 = 0 + 푟 = 푟 We have just proven the remainder theorem, which is formally stated in the box below. REMAINDER THEOREM: Let 푃 be a polynomial function in 푥, and let 푎 be any real number. Then there exists a unique polynomial function 푞 such that the equation 푃(푥) = 푞(푥)(푥 − 푎) + 푃(푎) is true for all 푥. That is, when a polynomial is divided by (푥 − 푎), the remainder is the value of the polynomial evaluated at 푎. Restate the remainder theorem in your own words to your partner. Lesson 19: The Remainder Theorem 207 This work is licensed under a This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from ALG II-M1-TE-1.3.0-07.2015 Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 19 M1 ALGEBRA II While students are doing this, circulate and informally assess student understanding before asking students to share their responses as a class. Exercise 4 (5 minutes) Students may need more guidance through this exercise, but allow them to struggle with Scaffolding: it first. After a few students have found 푘, share various methods used. Challenge early finishers with this problem: Exercise 4–6 Given that 푥 + 1 and 푥 − 1 are 4. Consider the polynomial 푷(풙) = 풙ퟑ + 풌풙ퟐ + 풙 + ퟔ. factors of 푃(푥) = 푥4 + 2푥3 − a. Find the value of 풌 so that 풙 + ퟏ is a factor of 푷. 49푥2 − 2푥 + 48, write 푃 in In order for 풙 + ퟏ to be a factor of 푷, the remainder must be zero. Hence, since factored form. 풙 + ퟏ = 풙 − (−ퟏ), we must have 푷(−ퟏ) = ퟎ so that ퟎ = −ퟏ + 풌 − ퟏ + ퟔ. Then 풌 = −ퟒ. Answer: (푥 + 1)(푥 − 1)(푥 + 8)(푥 − 6) b. Find the other two factors of 푷 for the value of 풌 found in part (a). 푷(풙) = (풙 + ퟏ)(풙ퟐ − ퟓ풙 + ퟔ) = (풙 + ퟏ)(풙 − ퟐ)(풙 − ퟑ) Discussion (7 minutes) . Remember that for any polynomial function 푃 and real number 푎, the remainder theorem says that there exists a polynomial 푞 so that 푃(푥) = 푞(푥)(푥 − 푎) + 푃(푎). What does it mean if 푎 is a zero of a polynomial 푃? 푃(푎) = 0 . So what does the remainder theorem say if 푎 is a zero of 푃? There is a polynomial 푞 so that 푃(푥) = 푞(푥)(푥 − 푎) + 0. How does (푥 − 푎) relate to 푃 if 푎 is a zero of 푃? If 푎 is a zero of 푃, then (푥 − 푎) is a factor of 푃. How does the graph of a polynomial function 푦 = 푃(푥) correspond to the equation of the polynomial 푃? The zeros are the 푥-intercepts of the graph of 푃. If we know a zero of 푃, then we know a factor of 푃. If we know all of the zeros of a polynomial function, and their multiplicities, do we know the equation of the function? Not necessarily. It is possible that the equation of the function contains some factors that cannot factor into linear terms. We have just proven the factor theorem, which is a direct consequence of the remainder theorem. FACTOR THEOREM: Let 푃 be a polynomial function in 푥, and let 푎 be any real number. If 푎 is a zero of 푃 then (푥 − 푎) is a factor of 푃. Give an example of a polynomial function with zeros of multiplicity 2 at 1 and 3. 2 2 푃(푥) = (푥 − 1) (푥 − 3) Lesson 19: The Remainder Theorem 208 This work is licensed under a This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from ALG II-M1-TE-1.3.0-07.2015 Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 19 M1 ALGEBRA II . Give another example of a polynomial function with zeros of multiplicity 2 at 1 and 3. 푄(푥) = (푥 − 1)2(푥 − 3)2(푥2 + 1) or 푅(푥) = 4(푥 − 1)2(푥 − 3)2 . If we know the zeros of a polynomial, does the factor theorem tell us the exact formula for the polynomial? No. But, if we know the degree of the polynomial and the leading coefficient, we can often deduce the equation of the polynomial. Scaffolding: Exercise 5 (8 minutes) Encourage students who are As students work through this exercise, circulate the room to make sure students have struggling to work on part (a) made the connection between zeros, factors, and 푥-intercepts. Question students to see using two methods: if they can verbalize the ideas discussed in the prior exercise. by finding 푃(1), and . by dividing 푃 by 푥 − 1. 푷(풙) = 풙ퟒ + ퟑ풙ퟑ − ퟐퟖ풙ퟐ − ퟑퟔ풙 + ퟏퟒퟒ 5. Consider the polynomial . This helps to reinforce the a. Is ퟏ a zero of the polynomial 푷? ideas discussed in Exercises 1 No and 2. b. Is 풙 + ퟑ one of the factors of 푷? Yes; 푷(−ퟑ) = ퟖퟏ − ퟖퟏ − ퟐퟓퟐ + ퟏퟎퟖ + ퟏퟒퟒ = ퟎ. c. The graph of 푷 is shown to the right. What are the zeros of 푷? Approximately −ퟔ, −ퟑ, ퟐ, and ퟒ. d. Write the equation of 푷 in factored form. 푷(풙) = (풙 + ퟔ)(풙 + ퟑ)(풙 − ퟐ)(풙 − ퟒ) . Is 1 a zero of the polynomial 푃? How do you know? No.