Lesson 19 NYS COMMON CORE MATHEMATICS CURRICULUM M1 ALGEBRA II Lesson 19: The Remainder Theorem Classwork Exercises 1β3 1. Consider the polynomial function π(π₯) = 3π₯ 2 + 8π₯ β 4. a. 2. b. Find π(2). Consider the polynomial function π(π₯) = π₯ 3 β 3π₯ 2 + 6π₯ + 8. a. 3. Divide π by π₯ β 2. Divide π by π₯ + 1. b. Find π(β1). b. Find β(3). Consider the polynomial function β(π₯) = π₯ 3 + 2π₯ β 3. a. Divide β by π₯ β 3. Lesson 19: Date: The Remainder Theorem 7/29/17 © 2014 Common Core, Inc. Some rights reserved. commoncore.org S.93 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 19 NYS COMMON CORE MATHEMATICS CURRICULUM M1 ALGEBRA II Exercises 4β6 4. 5. Consider the polynomial π(π₯) = π₯ 3 + ππ₯ 2 + π₯ + 6. a. Find the value of π so that π₯ + 1 is a factor of π. b. Find the other two factors of π for the value of π found in part (a). Consider the polynomial π(π₯) = π₯ 4 + 3π₯ 3 β 28π₯ 2 β 36π₯ + 144. a. Is 1 a zero of the polynomial π? b. Is π₯ + 3 one of the factors of π? c. The graph of π is shown to the right. What are the zeros of π? d. Write the equation of π in factored form. Lesson 19: Date: The Remainder Theorem 7/29/17 © 2014 Common Core, Inc. Some rights reserved. commoncore.org S.94 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 19 NYS COMMON CORE MATHEMATICS CURRICULUM M1 ALGEBRA II 6. Consider the graph of a degree 5 polynomial shown to the right, with π₯-intercepts β4, β2, 1, 3, and 5. a. Write a formula for a possible polynomial function that the graph represents using π as constant factor. b. Suppose the π¦-intercept is β4. Adjust your function to fit the π¦-intercept by finding the constant factor π. Lesson 19: Date: The Remainder Theorem 7/29/17 © 2014 Common Core, Inc. Some rights reserved. commoncore.org S.95 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 19 NYS COMMON CORE MATHEMATICS CURRICULUM M1 ALGEBRA II Lesson Summary Remainder Theorem: Let π be a polynomial function in π₯, and let π be any real number. Then there exists a unique polynomial function π such that the equation π(π₯) = π(π₯)(π₯ β π) + π(π) is true for all π₯. That is, when a polynomial is divided by (π₯ β π), the remainder is the value of the polynomial evaluated at π. Factor Theorem: Let π be a polynomial function in π₯, and let π be any real number. If π is a zero of π, then (π₯ β π) is a factor of π. Example: If π(π₯) = π₯ 2 β 3 and π = 4, then π(π₯) = (π₯ + 4)(π₯ β 4) + 13 where π(π₯) = π₯ + 4 and π(4) = 13. Example: If π(π₯) = π₯ 3 β 5π₯ 2 + 3π₯ + 9, then π(3) = 27 β 45 + 9 + 9 = 0, so (π₯ β 3) is a factor of π. Problem Set 1. Use the Remainder Theorem to find the remainder for each of the following divisions. a. (π₯ 2 + 3π₯ + 1) ÷ (π₯ + 2) b. (π₯ 3 β 6π₯ 2 β 7π₯ + 9) ÷ (π₯ β 3) c. (32π₯ 4 + 24π₯ 3 β 12π₯ 2 + 2π₯ + 1) ÷ (π₯ + 1) d. (32π₯ 4 + 24π₯ 3 β 12π₯ 2 + 2π₯ + 1) ÷ (2π₯ β 1) e. Hint for part (d): Can you rewrite the division expression so that the divisor is in the form (π₯ β π) for some constant π? 2. Consider the polynomial π(π₯) = π₯ 3 + 6π₯ 2 β 8π₯ β 1. Find π(4) in two ways. 3. Consider the polynomial function π(π₯) = 2π₯ 4 + 3π₯ 2 + 12. 4. 5. a. Divide π by π₯ + 2 and rewrite π in the form (divisor)(quotient) + remainder. b. Find π(β2). Consider the polynomial function π(π₯) = π₯ 3 + 42. a. Divide π by π₯ β 4 and rewrite π in the form (divisor)(quotient) + remainder. b. Find π(4). Explain why for a polynomial function π, π(π) is equal to the remainder of the quotient of π and π₯ β π. Lesson 19: Date: The Remainder Theorem 7/29/17 © 2014 Common Core, Inc. Some rights reserved. commoncore.org S.96 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 19 NYS COMMON CORE MATHEMATICS CURRICULUM M1 ALGEBRA II 6. Is π₯ β 5 a factor of the function π(π₯) = π₯ 3 + π₯ 2 β 27π₯ β 15? Show work supporting your answer. 7. Is π₯ + 1 a factor of the function π(π₯) = 2π₯ 5 β 4π₯ 4 + 9π₯ 3 β π₯ + 13? Show work supporting your answer. 8. A polynomial function π has zeros of 2, 2, β3, β3, β3, and 4. Find a possible formula for π and state its degree. Why is the degree of the polynomial not 3? 9. Consider the polynomial function π(π₯) = π₯ 3 β 8π₯ 2 β 29π₯ + 180. a. Verify that π(9) = 0. Since π(9) = 0, what must one of the factors of π be? b. Find the remaining two factors of π. c. State the zeros of π. d. Sketch the graph of π. Lesson 19: Date: The Remainder Theorem 7/29/17 © 2014 Common Core, Inc. Some rights reserved. commoncore.org S.97 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 19 NYS COMMON CORE MATHEMATICS CURRICULUM M1 ALGEBRA II 10. Consider the polynomial function π(π₯) = 2π₯ 3 + 3π₯ 2 β 2π₯ β 3. a. Verify that π(β1) = 0. Since π(β1) = 0, what must one of the factors of π be? b. Find the remaining two factors of π. c. State the zeros of π. d. Sketch the graph of π. 11. The graph to the right is of a third degree polynomial function π. a. State the zeros of π. b. Write a formula for π in factored form using π for the constant factor. c. Use the fact that π(β4) = β54 to find the constant factor. d. Verify your equation by using the fact that π(1) = 11. Lesson 19: Date: The Remainder Theorem 7/29/17 © 2014 Common Core, Inc. Some rights reserved. commoncore.org S.98 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 19 NYS COMMON CORE MATHEMATICS CURRICULUM M1 ALGEBRA II 12. Find the value of π so that (π₯ 3 β ππ₯ 2 + 2) ÷ (π₯ β 1) has remainder 8. 13. Find the value π so that (ππ₯ 3 + π₯ β π) ÷ (π₯ + 2) has remainder 16. 14. Show that π₯ 51 β 21π₯ + 20 is divisible by π₯ β 1. 15. Show that π₯ + 1 is a factor of 19π₯ 42 + 18π₯ β 1. Write a polynomial function that meets the stated conditions. 16. The zeros are β2 and 1. 17. The zeros are β1, 2, and 7. 1 2 3 4 18. The zeros are β and . 2 3 19. The zeros are β and 5, and the constant term of the polynomial is β10. 3 2 20. The zeros are 2 and β , the polynomial has degree 3 and there are no other zeros. Lesson 19: Date: The Remainder Theorem 7/29/17 © 2014 Common Core, Inc. Some rights reserved. commoncore.org S.99 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
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