M3 Topic 1 Student Packet-1 PDF

Test will be on: ___________________________ M3 Topic 1 Test Objectives Students will be able to: Packet pg A. Factor higher degreed polynomials over the set of Real Numbers B. Factor polynomials over the set of Complex Numbers (factors may be Real or Imaginary) C. Identify zeros of polynomials if given a factored form function D. Determine factors of a function through polynomial long division and quadratic formula E. Use the Remainder Theorom to evaluate polynomial equation and functions F. Use the Factor Theorem to determine an unknown coefficient in a polynomial function G. Identify if polynomials are closed under a given operation H. Determine solutions to a polynomial inequality How do I prepare for this test? It is important to actively participate in class each day and ask questions when you don’t understand something. When a quiz is returned, rework all problems you got incorrect and make sure you understand what you did wrong and know how to get the correct answer. Read through your notes! Find questions in the text, skills practice, or on worksheets provided throughout this lesson that helps you practice each objective. Check answers using Keys. pg 1 1 Satisfactory Factoring Relating Factors and Zeros Warm Up Learning Goals Solve each equation for x. Factor higher order polynomials. Distinguish between factoring polynomial equations 1. 2x2 2 4 5 8 over the set of real numbers and over the set of 2. (x 2 1)3 2 5 5 0 complex numbers. Identify zeros of polynomials when suitable factorizations 3. 3(x 2 6) 4 1 11 5 15 are available. Use the zeros of a polynomial to sketch a graph of the 4. x3 2 27 5 0 function. © Carnegie Learning, Inc. You have determined factors of degree-2 equations. How can you factor higher-degree polynomial functions? LESSON 1: Satisfactory Factoring M3-7 pg 2 GETTING STARTED Factor Tree Factory 24 At the Factor Tree Factory, a factor machine takes any whole number as input and outputs one of its factor pairs. 1. Suppose the number 24 is entered into the machine. ? a. What factor pairs might you see as the output? b. How do you know whether two numbers are a factor pair of 24? c. Can 5 be an output value? Explain your reasoning. 2. Cherise and Jemma each begin a factor tree for 24 using different outputs from the factor machine. Cherise Jemma 24 24 2 12 3 8 Cherise says both factor trees will show the same prime factorization when completed. Jemma says because © Carnegie Learning, Inc. they each started with a different factor pair, the prime factorizations will be different. Who’s correct? Complete each factor tree to justify your answer. M3-8 TOPIC 1: Relating Factors and Zeros 3. Consider the expression 2 ? 2 ? 2 ? 3. a. How does 2 ? 2 ? 2 ? 3 relate to the factors determined by Cherise and Jemma? b. How does it relate to 24? 4. If you know a factor of a given whole number, how can you determine another factor? 5. What is the remainder when you divide a whole number by any of its factors? Explain your reasoning. 6. Create a factor tree to show the prime factorization of © Carnegie Learning, Inc. each number. a. 66 b. 210 LESSON 1: Satisfactory Factoring M3-9 pg 4 AC T I V I T Y Factoring Out a GCF 1.1 Throughout your previous mathematics courses, you have applied the idea that a whole number can be decomposed into a product of its factors to solve a variety of problems. In this lesson, you will explore different methods of decomposing a polynomial into a product of its factors. Once you have factored a polynomial, you can use the factors to identify the zeros and then use the zeros to sketch a graph. To begin factoring any polynomial, always look for a greatest common factor (GCF). You can factor out the greatest common factor of the polynomial, and then factor what remains. 1. Ping and Shalisha each attempt to factor 3x3 1 12x2 2 36x by factoring out the greatest common factor. Ping Shalisha 3x3 1 12x2 2 36x 3x3 1 12x2 2 36x 3x(x2 1 4x 2 12) 3(x3 1 4x2 2 12x) Remember: a. Analyze each student’s work. Determine which student is correct and explain the inaccuracy in the other student’s work. © Carnegie Learning, Inc. A greatest common factor can be a variable, constant, or both. b. Completely factor the expression that Ping and Shalisha started to factor. y c. Use the factors to identify the zeros of f(x) = 3x3 1 12x2 2 36x. Then sketch the graph of the polynomial. x M3-10 TOPIC 1: Relating Factors and Zeros pg 5 2. Factor each polynomial function and identify the zeros. Then, use the factors to sketch a graph of the function defined by the polynomial. Remember: y a. f(x) = 3x3 1 3x2 2 6x Look for a greatest common factor first. Identify the zeros and other key points x before graphing. b. f(x) = 2x2 1 6x c. f(x) = 3x2 2 3x 2 6 y y x x d. f(x) = 10x2 2 50x 2 60 3. Analyze the factored form and the y corresponding graphs in Questions 1 and 2. What do the graphs in Question © Carnegie Learning, Inc. 1 and Question 2, parts (a) and (b) have in common that the graphs of Question x 2, parts (c) and (d) do not? Explain your reasoning. 4. Write a statement about the graphs of all polynomials that have a monomial GCF that contains a variable. LESSON 1: Satisfactory Factoring M3-11 pg 6 5. Tony and Eva each attempt to factor f(x) = x3 − 2x2 + 2x. Analyze their work. Tony First, I removed the GCF, x. The expression x2 - 2x + 2 cannot be factored, so f(x) = x(x2 - 2x + 2). Eva f(x) = x(x2 - 2x + 2) x2 − 2x + 2 _________________ = 0 2(22) ± x = _________________________ √ (22) 2 4(1)(2) 2 2(1) _____ 2 ± √2 4 x = __________ 2 2 ± 2i x = _________________________ 2 x = 1 ± i The function in factored form is f(x) = (x)[x − (1 + i)][x − (1 − i)]. a. If you consider the set of real numbers, who’s correct? If you consider the set of complex numbers, who’s correct? Explain your reasoning. © Carnegie Learning, Inc. b. Use the Distributive Property to rewrite Eva’s function to verify that the function in factored form is equivalent the Remember: original function in standard form. The set of complex c. Identify the zeros of the function f(x). numbers is the set of numbers that includes both real and imaginary numbers. M3-12 TOPIC 1: Relating Factors and Zeros pg 7 6. Analyze each expression. x2 + 4 x2 − 4 x2 + 2x + 5 x2 + 4x − 5 Some functions can be factored over the −x2 + x + 12 x2 + 4x − 1 −x2 + 6x − 25 set of real numbers. However, all functions can be factored over the set of complex a. Sort each expression based on whether it can be numbers. factored over the set of real numbers or over the set of imaginary numbers. Complex Factors Real Factors Imaginary Factors © Carnegie Learning, Inc. b. Factor each expression over the set of complex numbers. LESSON 1: Satisfactory Factoring M3-13 pg 8 AC T I V I T Y Using Structure to 1.2 Factor Polynomials Certain polynomials in quadratic form may have common factors in some of the terms, but not all terms. In this case, it may be helpful to write the terms as a product of 2 terms. You can then substitute the common term with a variable, z, and factor as you would any polynomial in quadratic form. This method of factoring is called chunking. Worked Example You can use chunking to factor 9x2 1 21x 1 10. Notice that the first and second terms both contain the common factor 3x. 9x2 1 21x 1 10 5 (3x)2 1 7(3x) 1 10 Rewrite terms as a product of common factors. 5 z2 1 7z 1 10 Let z 5 3x. 5 (z 1 5)(z 1 2) Factor the quadratic. 5 (3x 1 5)(3x 1 2) Substitute 3x for z. The factored form of 9x2 1 21x 1 10 is (3x 1 5)(3x 1 2). 1. Use chunking to factor and identify the zeros of f(x) 5 25x2 1 20x 2 21. Then sketch the polynomial. © Carnegie Learning, Inc. y x M3-14 TOPIC 1: Relating Factors and Zeros pg 9 2. Given z2 1 2z 2 15 = (z 2 3)(z 1 5), write another polynomial in general form that has a factored form of (z 2 3)(z 1 5) with different values for z. A special form of a polynomial is a perfect square trinomial. A perfect square trinomial has first and last terms that are perfect squares and a Remember: middle term that is equivalent to 2 times the product of the first and last term's square root. Factoring a perfect square trinomial can occur in two forms. You can use the a2 2 2ab 1 b2 5 (a 2 b)2 difference of two a2 1 2ab 1 b2 5 (a 1 b)2 squares to factor a binomial of the form a2 2 b2. 3. Determine which of the polynomial expression(s) is a perfect The binomial a2 2 b2 square trinomial and write it as the square of a sum or 5 (a 1 b)(a 2 b). difference. If it is not a perfect square trinomial, explain why not. a. x4 1 14x2 2 49 b. 16x2 2 40x 1 100 c. 64x2 2 32x 1 4 d. 9x4 1 6x2 1 1 © Carnegie Learning, Inc. LESSON 1: Satisfactory Factoring M3-15 pg 10 In polynomials of 4 terms, you may notice that although not all terms share a common factor, pairs of terms might share a common factor. In this situation, you can factor by grouping. 4. Colt factors the polynomial expression x3 1 3x2 2 x 2 3. Colt x3 1 3x2 2 x 2 3 x2(x 1 3) 2 1(x 1 3) (x 1 3)(x2 2 1) (x 1 3)(x 1 1)(x 2 1) a. Explain the steps Colt took to factor the polynomial expression. x3 1 3x2 2 x 2 3 x2(x 1 3) 2 1(x 1 3) Step 1: (x 1 3)(x2 2 1) Step 2: (x 1 3)(x 1 1)(x 2 1) Step 3: © Carnegie Learning, Inc. b. Use the factors to identify the zeros of f(x) 5 x3 1 3x2 2 x 2 3 and then sketch the graph. y x M3-16 TOPIC 1: Relating Factors and Zeros pg 11 5. Use factor by grouping to factor and identify the zeros of f(x) 5 x3 1 7x2 2 4x 2 28. Then sketch the polynomial. y x 6. Braxton and Kenny both factor the polynomial expression x3 1 2x2 1 4x 1 8. Analyze the set of factors in each student’s work. Describe the set of numbers over which each student factored. According to the Braxton Kenny Fundamental Theorem of Algebra, x3 1 2x2 1 4x 1 8 x3 1 2x2 1 4x 1 8 any polynomial x2(x 1 2) 1 4(x 1 2) x2(x 1 2) 1 4(x 1 2) function of degree n must have exactly n (x2 1 4)(x 1 2) (x2 1 4)(x 1 2) complex factors: f(x) 5 (x 2 r1)(x 2 r2) … (x 1 2i)(x 2 2i)(x 1 2) (x 2 rn) where r ∈ {complex numbers}. © Carnegie Learning, Inc. LESSON 1: Satisfactory Factoring M3-17 pg 12 Some degree-4 polynomials, written as a trinomial ax4 1 bx2 1 c, have the same structure as quadratics. When this is the case, the polynomial may be factored using the same methods you would use to factor a quadratic. This is called factoring using quadratic form. Worked Example Factor x4 2 29x2 1 100 using quadratic form. x4 2 29x2 1 100 Determine whether you can factor the given trinomial into 2 factors. (x2 2 4)(x2 2 25) Determine whether you can continue to factor each binomial. (x 2 2)(x 1 2)(x 2 5)(x 1 5) 7. Factor each polynomial over the set of complex numbers. Use the factors to identify the zeros and then sketch the polynomial. a. f(x) 5 x4 2 4x3 2 x2 1 4x b. f(x) 5 x4 2 10x2 1 9 y y x x © Carnegie Learning, Inc. M3-18 TOPIC 1: Relating Factors and Zeros pg 13 NOTES TALK the TALK Fracture It to Factor It You have used many different methods of factoring: Factoring Out the Greatest Common Factor Chunking Factoring by Grouping Perfect Square Trinomials Factoring Using Quadratic Form Depending on the polynomial, some methods of factoring are more eﬃcient than others. 1. Complete the table on the next page by matching each polynomial with the method of factoring you would use from the bulleted list given. Every method from the bulleted list should be used only once. Explain why you chose the factoring method for each polynomial. Finally, write the polynomial in factored form over the set of real numbers. © Carnegie Learning, Inc. LESSON 1: Satisfactory Factoring M3-19 pg 14 Method of Polynomial Reason Factored Form Factoring 3x4 1 2x2 2 8 x2 2 12x 1 36 x3 1 2x2 1 7x 1 14 25x2 2 30x 2 7 2x4 1 10x3 1 12x2 2. Factor each polynomial over the set of complex numbers. Explain why you chose the factoring method you used. a. x4 2 7x2 2 18 b. x4 1 3x2 2 28 © Carnegie Learning, Inc. M3-20 TOPIC 1: Relating Factors and Zeros pg 15 Assignment Write Remember Describe the similarity between the You can factor out the GCF of a polynomial and then factor chunking method of factoring and what remains. Analyzing the structure of a polynomial can factoring by grouping. Discuss what help you decide the most eﬃcient method for factoring. Once the structure of a polynomial would you have factored a polynomial, you can use the factors to look like in order for you to consider identify the zeros and then use the zeros to sketch a graph. using each method. Practice 1. Factor each polynomial over the set of real numbers. Use the factors to sketch the polynomial. a. f(x) 5 25x2 2 10x 2 24 b. f(x) 5 x3 2 4x2 2 9x 1 36 c. f(x) 5 x 4 2 25x2 1 144 d. f(x) 5 27x3 2 18x2 1 3x e. f(x) 5 16x3 1 54 f. f(x) 5 7x4 2 56x Stretch 1. Sketch each piecewise function. x, x , 21 { a. g(x) 5 { 2x 1 1, x , 0 b. f(x) 5 x 1 x 2 x 2 1, 21 # x # 1 3 2 2x2 2 8x x $ 0 4 x.1 y y © Carnegie Learning, Inc. 8 8 6 6 4 4 2 2 −8 −6 −4 −2 0 2 4 6 8 x −8 −6 −4 −2 0 2 4 6 8 x −2 −2 −4 −4 −6 −6 −8 −8 LESSON 1: Satisfactory Factoring M3-21 pg Name:____________________________________________ Date:________________Period:_________ 16 Notes: Factoring – Factor a Sum or Difference of two cubes Factor a sum or difference of cubes A sum or difference of cubes factors as follows: sum of cubes a3 + b3 = (a + b)(a2 – ab + b2) difference of cubes a3 – b3 = (a – b)(a2 + ab + b2) SOFAS is an acronym to help us remember Do you remember how to factor a sum or difference of cubes. your perfect cubes? S– x 1 x3 1 2 8 O– 3 27 4 64 F– 5 125 6 216 A– 7 343 8 512 S– 9 729 10 1,000 Examples: 1. 𝑥 3 − 64 2. 𝑥 3 + 125 3. 27𝑥 3 − 1000 4. 8𝑥 3 + 1 © mandy’s math world 2021 Name:____________________________________________ Date:________________Period:_________pg 17 Practice A2.___ Factoring – Factor a Sum or Difference of two cubes Factor each polynomial by factoring a sum or difference of two cubes. Be sure to factor out a GCF first, if necessary. 1. 𝑥 3 − 8 2. 𝑦 3 + 512 3. 𝑎3 + 1 4. 𝑚3 − 729 5. 512𝑥 3 − 1 6. 8𝑥 3 + 27 7. 3𝑥 4 − 3𝑥 8. 64𝑥 3 + 125 9. 𝑥 3 − 343𝑦 3 10. 32𝑥 3 − 4 pg 18 GETTING STARTED The x – r Factor You have analyzed the graphs of polynomials to determine the type and location of the zeros. How can you determine the factors of a polynomial given its algebraic representation and one of its factors? 1. Analyze the graph of the function h(x) = x3 + 12x2 + 41x + 72. y h(x) 80 60 40 20 –8 –6 –4 –2 0 2 4 6 8 x –20 –40 –60 –80 a. Describe the key characteristics of h(x). © Carnegie Learning, Inc. b. Describe the number and types of zeros of h(x). The Factor Theorem states that a polynomial function p(x) has (x 2 r) as a factor if and only if the value of the function at r is 0, or p(r) 5 0. M3-24 TOPIC 1: Relating Factors and Zeros pg 19 You can use the Factor Theorem to show that a linear expression is a factor of a polynomial. Worked Example Consider the graph of the polynomial function h(x) 5 x3 1 12x2 1 41x 1 72 in Question 1. The graph appears to have a zero at (28, 0), so a possible linear factor of the polynomial is (x 1 8). Determine the value of the polynomial at x 5 28, or h(28). h(28) 5 (28)3 1 12(-8)2 1 41(28) 1 72 5 2512 1 768 1 (2328) 1 72 50 So, (x 1 8) is a linear factor of the polynomial function. 2. Consider that d(x) = (x + 8), and d(x) · q(x) = h(x). a. What do you know about the function q(x)? © Carnegie Learning, Inc. b. Can you write the algebraic representation for q(x)? Explain your reasoning. LESSON 2: Divide and Conquer M3-25 pg 20 AC T I V I T Y Polynomial Long Division 2.1 To solve 0 5 x3 1 12x2 1 41x 1 72, you need to factor the polynomial and use the Zero Product Property to determine its zeros. The Fundamental Theorem of Algebra states that every polynomial equation of degree n must have n roots. This means that every polynomial Remember: can be written as the product of n factors of the form (ax 1 b). For example, 2x2 2 3x 2 9 5 (2x 1 3)(x 2 3). Recall that a 4 b is __ a , If 2 is a factor of 24, then 24 can be divided by 2 without a remainder. In the b where b Þ 0. same way, the factors of a polynomial divide into that polynomial without a remainder. Polynomial long division is an algorithm for dividing one polynomial by another of equal or lesser degree. The process is similar to integer long division. Worked Example Integer Long Division Polynomial Long Division x 3 1 12x 2 1 41x 1 72 _____________________ 3660 4 12 (x3 1 12x2 1 41x 1 72) 4 (x 1 8) or x+28 A D G or x2 1 4x 1 9 x3 3660 _____ ___________________ A. Divide __ x 5x. 2 x 1 8 Q x3 1 12x2 1 41x 1 72 © Carnegie Learning, Inc. 12 B B. Multiply x2(x 1 8), and then 2(x3 1 8x2) C 305 ______ E 4x2 1 41x subtract. Q 12 3660 C. Bring down 41x. 2(4x2 1 32x) 4x2 236 9x 1 72 F D. Divide ____ x 5 4x. H 6 E. Multiply 4x(x 1 8), and then 2(9x 1 72) 20 subtract. Remainder 0 60 F. Bring down 172. 9x G. Divide ___ 260 x 5 9. 0 H. Multiply 9(x 1 8), and then subtract. M3-26 TOPIC 1: Relating Factors and Zeros Long Division Examples: ! ! "#!"$$ A. !"% &! ! '#!"() B. (!'* (! " '( C. !'+ ! " "(! ! "#* D. !"* pg 22 1. Analyze the worked example that shows integer long division and polynomial long division. a. In what ways are the integer and polynomial long division algorithms similar? b. Rewrite each expression as a product of its factors. 3660 = ? x3 + 12x2 + 41x + 72 = ? c. Is h(x) completely factored? Explain your reasoning. d. Rewrite the function as a product of its linear factors. e. Determine the zeros of the function. © Carnegie Learning, Inc. 2. The expression (x − 7) is a factor of x3 − 10x2 + 11x + 70. Solve 0 = x3 − 10x2 + 11x + 70 over the set of complex numbers. LESSON 2: Divide and Conquer M3-27 pg 23 AC T I V I T Y Factoring Special Binomials 2.2 Recall that you can use the difference of squares to factor a binomial of the form a2 2 b2. The binomial a2 2 b2 5 (a 1 b)(a 2 b). 1. Use the difference of squares to factor each binomial over the set of real numbers. a. x2 2 64 b. x4 2 16 c. x8 2 1 d. x4 2 y4 Now let’s consider expressions composed of perfect cubes, such as f(x) 5 x3 2 8. 2. Consider Kingston’s and Toby’s work. Kingston Toby I can use the Properties of y I looked at the Equality to determine the factors. 8 graph of f(x) = x3 x3 2 8 5 0 and could se that x3 5 8 4 x = 2 is one of © Carnegie Learning, Inc. x52 f(x) 5 (x 2 2)(x 2 2)(x 2 2) 0 the zeros, but the –4 4 x or –4 other two zeros f(x) 5 (x 2 2)3 are imaginary. –8 a. Describe Kingston’s error. b. Use long division to factor over the set f(x) = x3 − 8 of real numbers. M3-28 TOPIC 1: Relating Factors and Zeros pg 24 You can rewrite the expression x3 2 27 as (x)3 2 (3)3, and x3 1 27 as (x)3 1 (3)3. When you factor sums and differences of cubes, there is a special factoring formula you can use, which is similar to the difference of squares for quadratics. To determine the formula for the difference of cubes, generalize the difference of cubes as a3 2 b3. Worked Example To determine the factor formula for the difference of cubes, factor out (a 2 b) by considering (a3 2 b3) 4 (a 2 b). a2 1 ab 1 b2 a 2 b )a3 2 0a2b 1 0ab2 2 b3 2(a3 2 a2b) a2b 1 0ab2 When performing long division, make 2(a2b 2 ab2) sure that the dividend ab2 2 b3 is in descending order. 2(ab2 2 b3) If any powers are not 0 included, use a zero to help with spacing. Therefore, the difference of cubes can be rewritten in factored form as: a3 2 b3 5 (a 2 b)(a2 1 ab 1 b2). 3. Use Properties of Equality to determine one zero. Then factor each polynomial function over the set of real numbers. a. f(x) = x3 2 27 b. g(x) = x3 1 27 © Carnegie Learning, Inc. 4. Determine the formula for the sum of cubes by dividing a3 1 b3 by (a 1 b). 5. Use the sum or difference of cubes to factor each binomial over the set of real numbers. a. x3 1 125 b. 8x3 2 1 c. x6 2 8 d. x9 1 y9 LESSON 2: Divide and Conquer M3-29 pg 25 pg 26 You learned that the process of dividing polynomials is similar to the process of dividing integers. Sometimes when you divide two integers there is a remainder, and sometimes there is not a remainder. What does each case mean? In this activity, you will investigate what the remainder means in terms of polynomial division. 1. Use long division to determine the quotient for each. a. (" ! + 2" " − 5" + 16) ÷ (" − 4) Rewrite dividend as a product of Quotient Answer (divisor)(quotient) + remainder b. (4" # + 5" " − 7" + 9) ÷ (2" − 3) Rewrite dividend as a product of Quotient Answer (divisor)(quotient) + remainder c. (9" # + 3" ! + 4" " + 7" + 2) ÷ (3" + 2) Rewrite dividend as a product of Quotient Answer (divisor)(quotient) + remainder 2. Consider Question 1 (a-c) and answer the following: a. When there is a remainder, is the divisor a factor of the dividend? b. Describe the remainder when you divide a polynomial by one of its factors. pg pg 27 26 Remember from your experiences with division that: dividend _________ remainder divisor 5 quotient 1 __________ divisor or dividend 5 (divisor) (quotient) 1 remainder. It follows that any polynomial, p(x), can be written in the form: p(x) _________________ remainder linear expression 5 quotient 1 _________________ linear expression or p(x) 5 (linear expression) (quotient) 1 remainder. Generally, the linear expression is written in the form (x 2 r), the quotient is represented by q(x), and the remainder is represented by R. p(x) 5 (x 2 r) q(x) 1 R 3. Consider each dividend in Question 1 as a function, p(x). a. In part (a), evaluate p(x) for x = 4. 3 b. In part (b), evaluate p(x) for x = __ 2. © Carnegie Learning, Inc. 2 c. In part (c), evaluate p(x) for x = −__ 3. Remember: d. How does the remainder relate to the divisor in The Factor Theorem states that a each problem? polynomial has a linear polynomial as a factor if and only if the remainder is zero. 4. What conclusion can you make about any polynomial Therefore, if R 5 0, then f(r) 5 0, evaluated at r ? and (x 2 r) is a factor of f(x). LESSON 2: Divide and Conquer M3-31 pg 28 27 The Remainder Theorem states that when any polynomial function, f(x), is divided by a linear expression of the form (x 2 r), the remainder R 5 f(r), or the value of the function when x 5 r. p(x) _______ 5. Given p(x) = x3 + 6x2 + 5x − 12 and (x − 2) = x2 + 8x + 21 R 30, Rico says that p(−2) = 30 and Paloma says that p(2) = 30. Without performing any calculations, who is correct? Explain your reasoning. 6. The function f(x) = 4x2 + 2x + 9 generates the same remainder © Carnegie Learning, Inc. when divided by (x − r) and (x − 2r), where r is not equal to 0. Calculate the value(s) of r. M3-32 TOPIC 1: Relating Factors and Zeros pg 29 28 7. Determine the unknown in each. x a. __ 7 = 18 R 2. Determine x. p(x) b. ______ x + 3 = 3x + 14x + 15 R 3. Determine the function p(x). 2 c. Describe the similarities and differences in your solution strategies. © Carnegie Learning, Inc. LESSON 2: Divide and Conquer M3-33 pg 30 29 5. Determine the zeros of each function. a. f(x) = x3 + 11x2 + 37x + 42 y 40 20 –8 –6 –4 –2 0 2 x –20 –40 b. f(x) = x3 − 4.75x2 + 3.125x − 0.50 y 2 –2 0 2 4 6 x –2 –4 © Carnegie Learning, Inc. –6 LESSON 2: Divide and Conquer M3-37 pg 31 30 NOTES TALK the TALK A Polynomial Divided In the Getting Started, you learned that the Factor Theorem is a way to show that a linear expression is a factor of a polynomial. You can also use the Factor Theorem to determine unknown information about a polynomial if you know a linear factor of that polynomial. Worked Example Given the function f(x) 5 x3 2 4x2 2 ax 1 10. You can determine the unknown coefficient, a, given that x 2 5 is a linear factor. If (x 2 5) is a linear factor, then, by the Factor Theorem, f(5) 5 0. f(5) 5 53 2 4(5)2 2 5a 1 10 0 5 53 2 4(5)2 2 5a 1 10 0 5 125 2 100 2 5a 1 10 0 5 35 2 5a 5a 5 35 a57 So, the unknown coefficient, a, is equal to 7, and © Carnegie Learning, Inc. f(x) 5 x3 2 4x2 2 7x 1 10. 1. Use the worked example to determine the unknown coefficient, a, in each function. Then identify the zeros of the functions over the set of complex numbers. a. f(x) = x3 − 9x2 + ax + 60, if x − 5 is a linear factor. b. f(x) = x4 + ax2 − 3, if x − 1 is a linear factor. M3-38 TOPIC 1: Relating Factors and Zeros pg 32 31 2. Given the information, determine whether each statement is NOTES true or false. Explain your reasoning. p(x) 5 x3 1 6x2 1 11x 1 6, and p(x) 4 (x 1 4) 5 x2 1 2x 1 3 R 26 a. p(−4) = −6 b. p(x) = (x + 4)(x2 + 2x + 3) − 6 c. (x − 3) is a factor of p(x) d. (x + 2) is a factor of p(x) © Carnegie Learning, Inc. 3. Explain the difference between the Remainder Theorem and the Factor Theorem. LESSON 2: Divide and Conquer M3-39 pg 33 32 Assignment Write Remember Write an example for each term using the A polynomial function p(x) has (x 2 r) as a factor dividend x 2 2x 1 4 and the divisor x 1 1. 2 if and only if the value of the function at r is 0, 1. Factor Theorem or p(r) 5 0. When any polynomial equation or 2. Polynomial long division function f(x) is divided by a linear expression of 3. Remainder Theorem the form (x 2 r) , the remainder is R 5 f(r) or the 4. Synthetic division value of the equation or function when x 5 r. Practice 1. Use the Factor Theorem to determine whether each linear expression is a factor of the polynomial x4 1 x3 2 17x2 1 15x. a. x 1 3 b. x 1 5 c. x 2 1 2. Factor each binomial over the set of real numbers. a. 4x2 2 9y2 b. x3 1 216 3. The Polynomial Pool Company offers 10 different pool designs numbered 1 through 10. Each pool is in the shape of a rectangular prism. The volume of water in Pool Design x can be determined using the function V(x) 5 l(x) ? w(x) ? d(x) 5 2x3 1 18x2 1 46x 1 30, where l(x), w(x), and d(x) represent the length, width, and depth of the pool in feet. a. Determine the expressions for the functions w(x) and d(x) if l(x) 5 2x 1 2 and the width of each pool is greater than the depth. Do not use a calculator. b. Determine the length, width, and depth of Pool Design 9. 4. The function m(x) 5 2x2 1 6x 2 7 generates the same remainder when divided by (x 2 a) and (x 2 2a) when a fi 0. Calculate the value(s) of a and determine the corresponding factors. 5. The given table of values represents the function f(x) 5 x3 1 9x2 1 14x 2 24. © Carnegie Learning, Inc. (x) 22 21 0 1 2 f(x) 224 230 224 0 48 a. Determine one of the factors of f(x) without using a calculator. Explain your reasoning. b. Completely factor f(x) without using a calculator. c. Determine all of the zeros of f(x) without using a calculator. LESSON 2: Divide and Conquer M3-41 3 Closing Time The Closure Property Warm Up Learning Goals Identify which of the functions are polynomials. Compare functions that are closed under Explain your reasoning. addition, subtraction, and multiplication 1. f(x) 5 6 to functions that are not closed under these operations. 1 2. g(x) 5 __x Analyze the meaning for polynomials to be __ closed under an operation. 3. h(x) 5 √ x Compare integer and polynomial operations. 4. m(x) 5 2 x 5. n(x) 5 x229 Key Term closed under an operation © Carnegie Learning, Inc. You have added, subtracted, multiplied, and divided two or more polynomial functions to build a new polynomial function. How are these operations similar to those involving integers? LESSON 3: Closing Time M3-43 GETTING STARTED Need for Closure Throughout this course, you have added, subtracted, multiplied, and divided two or more polynomial functions to build a new polynomial function. You did this using a graph, algebra, and a table of values. When an operation is performed with any numbers or expressions in a set and the result is in the same set, it is said to be closed under that operation. Are polynomials closed under addition, subtraction, multiplication, and division? In other words, when you add, subtract, multiply, or divide polynomial functions, do you always create another polynomial function? Before answering this question, let’s analyze closure within the real number system. 1. Determine whether each set within the real number system is closed under addition, subtraction, multiplication, and division. a. Complete the table. If a set is not closed under a given operation, provide a counterexample. Addition Subtraction Multiplication Division Natural Numbers {1, 2, 3, 4,...} Whole Numbers No Yes {0, 1, 2, 3,...} 2 2 3 5 21 © Carnegie Learning, Inc. Integers {... −2, −1, 0, 1, 2...} Rational Numbers Can be represented as the ratio of two integers Irrational Numbers Cannot be represented as the ratio of two integers b. What patterns do you notice? M3-44 TOPIC 1: Relating Factors and Zeros AC T I V I T Y The Closure Property 3.1 for Polynomials You conjectured that integers are closed under addition, subtraction, and multiplication. You also determined through counterexamples that integers are not closed under division. 1. Similarities between integer and polynomial operations are shown in the table. Integer Example Polynomial Example 400 1 30 1 7 4x2 1 3x 1 7 Addition 1 20 1 5 1 2x 1 5 400 1 50 1 12 4x2 1 5x 1 12 400 1 30 1 7 4x2 1 3x 1 7 Subtraction 2 (20 1 5) 2 (2x 1 5) 400 1 10 1 12 4x 1 x 1 2 2 400 1 30 1 7 4x2 1 3x 1 7 3 20 1 5 3 2x 1 5 Multiplication 2000 1 150 1 35 20x2 1 15x 1 35 8000 1 600 1 140 8x3 1 6x2 1 14x 8000 1 2600 1 290 1 35 8x3 1 26x2 1 29x 1 35 437 ____ 4x2 1 3x 1 7 ____________ Division 25 5 17 R12 2x 1 5 5 (2x 2 3) R(2x 1 22) © Carnegie Learning, Inc. a. Describe the similarities between polynomial and integer operations. b. In what ways is the Distributive Property essential to performing operations with integers and polynomials? LESSON 3: Closing Time M3-45 pg 37 4 Unequal Equals Solving Polynomial Inequalities Warm Up Learning Goals Consider the function f(x) 5 x 2 13x 1 36. 4 2 Represent problem situations using Identify the zeros and sketch a graph of the polynomial inequalities. polynomial. Determine solutions to polynomial inequalities algebraically and y graphically. © Carnegie Learning, Inc. x You have solved and graphed linear and quadratic inequalities. How can you graph inequalities involving polynomial expressions? LESSON 4: Unequal Equals M3-51 pg 38 2. Solve 18 ≤ 3x2 1 x. Show your work algebraically and graphically. y 16 12 8 4 –4 –3 –2 –1 0 1 2 3 4 x –4 –8 –12 –16 3. Solve each inequality and sketch a graph of the solution. a. 2x3 2 8x2 2 8x 1 32 > 0 y © Carnegie Learning, Inc. x M3-58 TOPIC 1: Relating Factors and Zeros pg 39 b. 6x3 2 21x2 2 12x > 0 y Think about: Consider the inequality sign x when graphing the polynomial. Will the graph be a dashed or solid smooth curve? c. x4 2 13x2 1 36 ≤ 0 y x © Carnegie Learning, Inc. LESSON 4: Unequal Equals M3-59 pg 40 pg 41 pg 42 pg 43 pg 44 pg 45 pg 46 pg 47 pg 48 pg 49 pg 50 pg 51

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