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Objective 5.1A New Vocabulary monomial degree of a monomial New Rules Rule for Multiplying Exponential Expressions Rule for Simplifying the Power of an Exponential Expression Rule for Simplifying Powers of Products Discuss the Concepts 1. Explain how to multiply two exponential expressions with the same base. 2. Why can’t the exponential expression x5y3 be simplified? 3. Which expression is the product of two exponential expressions and which is the power of an exponential expression? a. q4 ∙ q10 b. (q4)10 4. In the expression (a8b6)5, what is the product and what is the power? Concept Check 1. Find the area of the rectangle. The dimensions given are in kilometers. 54m2n4km2 9mn2 6mn2 2. Find the area of the square. The dimension given is in meters. 8x2y Optional Student Activity 1. Simplify: (5xy3)(3x4y2) – (2x3y)(x2y4) 2. Simplify: 4a2(2ab)3 – 5b2(a5b) 3. If 333 + 333 = 3x, find x. 34 13x5y5 27a5b3 64x4y2m2 Objective 5.1B New Definitions Definition of Zero as an Exponent Definition of a Negative Exponent New Rules Rule for Simplifying Powers of Quotients Rule for Negative Exponents on Fractional Expressions Rule for Dividing Exponential Expressions Discuss the Concepts 1. Explain how to divide two exponential expressions with the same base. 2. Why can’t the expression x8 be simplified? y2 3. Rewrite the following expressions with positive exponents. a. b-8 b. 1 w 5 Concept Check 1. The area of the rectangle below is 24a3b5 square yards. Find the length of the rectangle. 6ab4 yd 4a2b 2. The area of the parallelogram below is 56w4z6 square meters. Find the height of the parallelogram. 14w2z5 Optional Student Activity 1. The product of a monomial and 4b is 12a2b. Find the monomial. 2. The product of a monomial and 8y2 is 32x2y3. Find the monomial. 3. Simplify: 4. If 2040 4020 a a a aaa 1 3, find the value of a2. 9 3a2 4x2y 4w2z m Optional Student Activity Have your students complete the following equations and explain the pattern. 105 104 103 102 101 100,000; 10,000; 1000; 100; 10. The exponent on 10 decreases by 1 while each successive number is one-tenth of the number above it: 100,000 ÷ 10 = 10,000; 10,000 ÷ 10 = 1000; and so on. Now have them continue the established pattern. The next exponent on 10 is 1 – 1 = 0. 100 is equal to 10 ÷ 10 = 1. The next exponent is 0 – 1 = -1, and 10-1 is equal to 1 ÷ 10 = 0.1, and so on. The next eight equations are 100 = 1 10-1 = 0.1 10-2 = 0.01 10-3 = 0.001 10-4 = 0.0001 10-5 = 0.00001 10-6 = 0.000001 10-7 = 0.0000001 This exercise will enhance students’ understanding of x0 and of expressions with negative exponents. Objective 5.1C New Vocabulary scientific notation Discuss the Concepts 1. Name some situations in which scientific notation is used. Examples include molecular quantities and distances in the universe. 2. Determine whether the expression is written in scientific notation. a. 2.84 10-4Yes b. 36.5 10 No. 36.5 is not a number between 1 and 10. c. 0.91 10-1 No. 0.91 is not a number between 1 and 10. Concept Check Place the correct symbol, < or >, between the two numbers. 1. 5.23 1018 ? 5.23 1017 > 2. 3.12 1011 ? 3.12 1012 3. 3.45 10 -14 ? 3.45 10 -15 < > Note: You might extend this exercise by asking students to write a rule for ordering two numbers written in scientific notation: The number with the larger power of ten is the larger number. Objective 5.1D Optional Student Activity The rules for multiplying and dividing numbers written in scientific notation are the same as those for operating on algebraic expressions. The power of 10 corresponds to the variable expression, and the number between 1 and 10 corresponds to the coefficient of the variable. For example, (4x-3)(2x5) = 8x2 and (4 10-13)(2 105) = 8 102 or 6 x5 2 x7 3x 2 and 6 105 2 107 3 102 Simplify the following. 1. (1.9 1012)(3.5 107) 6.65 1019 2. (4.2 107)(1.8 10-5) 7.56 102 3. (2.3 10-8)(1.4 10-6) 3.22 10-14 4. 6.12 1014 1.7 109 5. 6 108 2.5 102 6. 5.58 107 3.11011 7. 9.03 106 4.3 105 3.6 105 2.4 10-6 1.8 10-18 2.1 1011 Answers to Writing Exercises 126. a. x0 = 1;The Definition of Zero as an Exponent was used incorrectly. b. (x4)5 = x4(5) = x20; The Rule for Simplifying the Power of an Exponential Expression was used incorrectly. c. x2 ∙ x3 = x2+3 = x5; The Rule for Multiplying Exponential Expressions was used incorrectly. Objective 5.2A Vocabulary to Review term [1.3B] monomial [5.1A] linear function [3.3A] evaluate a function [3.2A] constant term [1.3B] New Vocabulary polynomial binomial trinomial degree of a polynomial descending order polynomial function quadratic function cubic function leading coefficient Discuss the Concepts 1. State whether the polynomial is a monomial, a binomial, or a trinomial. Explain your answer. a. 8x4 6x2 b. 4a2b2 9ab 10 3 4 c. 7x y 2. State whether or not the expression is a polynomial. Explain your answer. a. 1 3 1 x x 5 2 b. 1 1 2 2x 5x c. x 5 Concept Check Determine whether the statement is always true, sometimes true, or never true. 1. The terms of a polynomial are monomials. Always true 2. The leading coefficient of 4 – 2x – 3x2 is 4. Never true 3. f(x) = 3x2 – 2x + x-1 is a polynomial function. 4. f(x) = x2 + 3x + 5x3 – 2 is a quadratic function. 5. A binomial is a polynomial of degree 2. Never true Never true Sometimes true 6. A cubic polynomial is a polynomial that contains three terms. Sometimes true Optional Student Activity 1. The height h, in feet, of a golf ball t seconds after it has been struck is given by h(t) = -16t2 + 60t. Determine the height of the ball 3 s after it is hit. 36 ft 2. Some forecasters believed that the revenue generated by business on the Internet around 2000 could be approximated by the function R(t) = 15.8t2 – 17.2t + 10.2, where R is the annual revenue in billions of dollars and t is the time in years, with t = 0 corresponding to the year 2000. Use this function to approximate the annual revenue in the year 2005. $319.2 billion 3. If $2000 is deposited into an individual retirement account (IRA), then the value, V, of that investment three years later is given by the cubic polynomial function V(r) = 2000r3 + 6000r + 2000, where r is the interest rate (as a decimal) earned on the investment. Determine the value after three years of $2000 deposited in an IRA that earns an interest rate of 7%. $2420.69 Objective 5.2B Vocabulary to Review additive inverse [1.1A] definition of subtraction [1.2A] New Vocabulary additive inverse of a polynomial Concept Check 1. Find the length of line segment AC given that the length of AB is 3x2 – 4x + 5 and the length of BC is 8x2 + 6x – 1. 11x2 + 2x + 4 A B C 2. The length of line segment LN is 7a2 + 4a -3. Given that the length of LM is 2a2 + a + 6, find the length of line segment MN. 5a2 + 3a – 9 L M N 3. Find the perimeter of the rectangle. The dimensions given are in kilometers. (8d2 + 12d + 4) km 3d2 + 5d -4 d2 + d + 6 Optional Student Activity A company’s revenue is the money the company earns by selling its products. A company’s cost is the money it spends to manufacture and sell its products. A company’s profit is the difference between its revenue and its cost. This relationship is expressed by the formula P = R – C where P is the profit, R is the revenue, and C is the cost. 1. A company manufactures and sells wood stoves. The total monthly cost, in dollars, to produce n wood stoves is 30n + 2000. The company’s revenue, in dollars, obtained from selling all n wood stoves is -0.4n2 + 150n. Express the company’s monthly profit in terms of n. (-0.4n2 + 120n – 2000) dollars 2. A company’s total monthly cost, in dollars, for manufacturing and selling n videotapes is 35n + 2000. The company’s revenue, in dollars, from selling all n videotapes is -0.2n2 + 175n. Express the company’s monthly profit in terms of n. (0.2n2 + 140n – 2000) dollars Answers to Writing Exercises 40. If P(x) is a third-degree polynomial and Q(x) is a fourth-degree polynomial, then P(x) + Q(x) is a fourth-degree polynomial. For example, let P(x) = 2x3 + 4x2 – 3x + 6 and let Q(x) = x4 – 5x + 1. Then P(x) + Q(x) = x4 + 2x3 + 4x2 – 8x + 7, a fourth-degree polynomial. 41. If P(x) is a fifth-degree polynomial and Q(x) is a fourth-degree polynomial, then P(x) – Q(x) is a fifth-degree polynomial. For example, let P(x) = 8x5 – x4 + 3x2 – 1 and let Q(x) = -x4 + x3 + x2 + 2x + 5. Then P(x) – Q(x) = 8x5 – x3 + 2x2 – 2x – 6, a fifth-degree polynomial. Objective 5.3A Vocabulary to Review monomial polynomial [5.1A] [5.2A] Properties to Review Distributive Property Concept Check 1. Find the area of a rectangle that has a length of 5x mi and a width of (2x – 7) mi. (10x2 – 35x) mi2 2. The base of a triangle is 4x m and the height is (2x + 5) m. Find the area of the triangle in terms of the variable x. (4x2 + 10x) m2 3. An athletic field has dimensions of 30 yd by 100 yd. An end zone that is w yards wide borders each end of the field. Express the total area of the field and the end zones in terms of the variable w. (60w + 3000) yd2 Optional Student Activity Have students explain why the following diagram represents (a + b)2 = a2 + 2ab + b2 a b a a2 ab b ab b2 Then ask them to draw diagrams to represent a. (x + 3)2 = x2 + 6x + 9 b. (y + 5)2 = y2 + 10y + 25 c. (x + y)2 = x2 + 2xy + y2 Objective 5.3B Vocabulary to Review term [1.3B] polynomial [5.2A] binomial [5.2A] New Vocabulary FOIL method Discuss the Concepts 1. When is the FOIL method used? 2. How is the Distributive Property used to multiply two polynomials? 3. If a polynomial of degree 3 is multiplied by a polynomial of degree 2, what is the degree of the resulting polynomial? Concept Check Determine whether the statement is always true, sometimes true, or never true. Always true 1. The product of two polynomials is a polynomial. 2. The FOIL method is used to multiply two polynomials. Sometimes true Sometimes true 3. The product of two binomials is a trinomial. 4. Using the FOIL method, the terms 3x and 5 are the “First” terms in (3x + 5)(2x + 7). Never true Optional Student Activity 1. Find all values of x that satisfy (3x2 + 5x – 2)(x + 3) = (3x2 + 8x – 3)(x + 2). 2. Find the coefficient of x5 in the product of x5 – 2x4 + 3x2 – x + 3 and x6 + 3x5 All real numbers – 4x3 + 2x2 + 3x – 4. 213 Objective 5.3C Vocabulary to Review FOIL method [5.3B] New Vocabulary product of the sum and difference of two terms square of a binomial Discuss the Concepts 1. What does it mean to square a binomial? 2. Why is (a + b)2 not equal to a2 + b2? Concept Check Simplify. 1. (a + b)2 – (a – b)2 2. (x + 3y)2 4ab + (x – 3y)(x – 3y) 2x2 + 18y2 Optional Student Activity 1. Find the product, in simplest form, of all the following binomials. x16 + 1 x8 + 1 x4 + 1 x2 + 1 x+1 x–1 x32 – 1 2. The squares of two consecutive positive integers differ by 1999. Find the sum of these two integers. x2, y2, z2 3. The numbers and are squares of consecutive positive integers, and x2 = 888, find the value of y. 222 Optional Student Activity 1. Multiply: (x + 1)(x – 1) x2 – 1 2. Multiply: (x + 1)(-x2 + x – 1) -x3 – 1 x2 < y2 < z2. 1999 Given that z2 – 3. Multiply: (x + 1)(x3 – x2 + x – 1) x4 – 1 4. Multiply: (x + 1) (-x4 + x3 – x2 + x – 1) -x6 – 1 5. Use the pattern of the answers to Exercises 1 to 4 to multiply x + 1 times x5 – x4 + x3 – x2 + x – 1. x6 – 1 6. -Use the pattern of the answers to Exercises 1 to 5 to multiply x + 1 times -x6 + x5 – x4 + x3 – x2 + x – 1. –x7 – 1 Objective 5.4A Vocabulary to Review polynomial monomial [5.2A] 5.1A] Discuss the Concepts 1. Describe two methods of simplifying the expression 2. Explain how to use multiplication to check that 12 36 . 6 8 x5 12 x3 equals 2x3 3x. 2 4x Concept Check 1. What is the quotient of 8x2y + 4xy and 2xy? 2. What is 6a2b2 – 9a2b + 18ab divided by 3ab? 4x + 2 2ab – 3a + 6 Optional Student Activity Divide. 1. 5 x 2 3x 1 x 2. 24 x 2 y 4 xy 8 x 2 2 xy 5x 3 1 x 12 x 2 4x y Objective 5.4B New Formulas Dividend (quotient divisor) + remainder Concept Check 1. Given that x3 1 x 2 x 1, name two factors of x3 1. x + 1 and x2 – x + 1 x 1 2. 3x + 1 is a factor of 3x3 – 8x2 – 33x – 10. Find a quadratic factor of 3x3 8x2 33x 10. x 12? x2 – 3x – 10 3. 4 x 1 is a factor of 8x3 38x2 49x 10. Find a quadratic factor of 8x3 38x2 49x 10. 2x2 – 9x + 10 4. Is 2 x 3 a factor of 4x3 x 12? Explain your answer. No. There is a remainder when 4x3 + x - 12 is divided by 2x – 3. Optional Student Activity 1. Divide: 3 x 2 xy 2 y 2 3x 2 y 2. Divide: 12 x 2 11xy 2 y 2 4x y 3. Divide: a 4 b4 ab x–y 3x + 2y a3 a2b ab2 b3 2b4 ab 4. When x x 2 is divided by a polynomial, the quotient is x 4 and the remainder is 14. Find the polynomial. x – 3 2 Optional Student Activity Divide each of the following polynomials by x y. 3 3 a. x y x2 + xy + y2 5 5 b. x y x4 + x3y + x2y2 + xy3 + y4 7 7 c. x y x6 + x5y + x4y2 + x3y3 + x2y4 + xy5 + y6 9 9 d. x y x8 + x7y + x6y2 + x6y3 + x4y4 + x3y5 + x2y6 + xy7 + y8 Explain the pattern and use the pattern to write the quotient of (x11 – y11) ÷ (x – y ). (x11 – y11) ÷ (x – y ) = x10 + x9y + x8y2 + x7y3 + x6y4 + x5y5 + x4y6 + x3y7 + x2y8 + xy9 + y10 Objective 5.4C Vocabulary to Review degree of a polynomial [5.2A] additive inverse [1.1A] binomial [5.2A] coefficient [1.3B] New Vocabulary synthetic division Discuss the Concepts 1. Suppose you are going to divide 2x3 13x2 15x 5 by x 5 using synthetic division. a. What are the coefficients of the dividend? b. What is the value of a? c. What is the degree of the first term of the quotient? 2. Why, in synthetic division, is addition used rather than subtraction? 3. When synthetic division is used to divide a polynomial by a binomial of the form x a, how is the degree of the quotient related to the degree of the dividend? 4. How can you check the answer to a synthetic division problem? Concept Check Which of the following divisions can be performed using synthetic division? a. (x2 + 3x + 1) ÷ (x – 2) Yes b. (x6 – 8x4 + 3x2 – 9) ÷ (x + 9) Yes c. (x4 – 5x3 – x2 + 7x + 3) ÷ (x2 + 1) No d. (x8 – 2) ÷ (x2 – 4) No e. (2x2 + 6x + 7) ÷ (5 – x) Yes Optional Student Activity 1. Two linear factors of x4 x3 7 x2 x 6 are x 1 and x 3. Find the other two linear factors of x4 x3 7 x2 x 6 . x – 2 and x + 1 2. A rectangular box has a volume of ( x3 11x 2 38 x 40) in 3 . The height of the box is x 2 in. Find the length and width of the box in terms of x. Length: (x + 5) in.; width: (x + 4) in. 3 2 3 3. The volume of a right circular cylinder is ( x 7 x 15 x 9) cm . The height of the cylinder is x 1 cm. Find the radius of the cylinder in terms of x. (x + 3) cm 4. When a polynomial P x is divided by a polynomial d x , it produces a quotient q x and a remainder r x . Either r x 0 or the degree of r x is less than the degree of the divisor d x . Why must the degree of r x be less than the degree of d x ? If the degree of r(x) is not less than the degree of d(x), then r(x) is divisible by d(x). Objective 5.4D Vocabulary to Review evaluating a polynomial function synthetic division [5.4C] [5.2A] New Vocabulary Remainder Theorem Discuss the Concepts 1. State the Remainder Theorem. 2. If the polynomial 3x4 8x2 2x 1 is divided by x 2 and the remainder is 13, what do we know about 4 f 2 for the function f x 3 x 8x2 2x 1 ? Optional Student Activity The Factor Theorem is a result of the Remainder Theorem. The Factor Theorem states that a polynomial P x has a factor x c if and only if P c 0. In other words, a remainder of zero means that the divisor is a factor of the dividend. 1. Determine whether x 5 is a factor of P x x4 x3 21x2 x 20. Yes 2. Based on your answer to Exercise 1, is 25 a zero of P x ? Explain your answer. Yes 3. Explain why P x 4 x4 7 x2 12 has no factor of the form x c , where c is a real number. The given polynomial has no factor of the form (x – c) because the value of the polynomial is always greater than 0 and thus never equal to 0. 4. Determine whether the second polynomial is a factor of the first. a. x3 8; x 2 Yes b. x3 8; x 2 No c. x 8; x 2 No d. x3 8; x 2 Yes 3 e. x 16; x 2 No f. x 4 16; x 2 Yes g. x 16; x 2 No h. x 4 16; x 2 Yes 4 4 Use your answers to parts (a) through (h) to make a conjecture as to whether the statement is true or false. n n i. For n > 0, x y is a factor of x y . True k. For n > 0 and n an odd integer, x y is a factor of x y . l. For n > 0 and n an even integer, x y is a factor of x y . m. For n > 0 and n an odd integer, x y is a factor of x y . n n j. For n > 0 and n an even integer, x y is a factor of x y . n n True False n n False n n True Answers to Writing Exercises 75. Synthetic division can be modified so that the divisor is of the form ax b. Divide both the dividend and the divisor by a (or multiply both the dividend and the divisor by x 1 ). The divisor is now in the form a b b , and the expression can be used for a in the divisor x – a of synthetic division. a a Objective 5.5A Vocabulary to Review monomial [5.1A] greatest common factor (GCF) [1.2B] New Vocabulary factor factoring a polynomial common monomial factor binomial factor Discuss the Concepts 1. Which of the following expressions are written in factored form? a. a3 4b 9 Yes 2 b. 2 y y 1 No c. 5c 6 c 8 Yes 2. Explain the meaning of “a factor” and the meaning of “to factor.” Concept Check Explain why the statement is true. 1. The terms of the binomial 3x 9 have a common factor. 2. The expression 3x2 15 is not in factored form. 3. 2 x 1 is a factor of x 2 x 1. Objective 5.5B Vocabulary to Review binomial factor [5.5A] common monomial factor [5.5A] New Vocabulary factoring by grouping Discuss the Concepts 1. Explain how you can rewrite b – a as – (a – b). 2. After factoring a polynomial by grouping, how can you check your answer? 3. Does grouping the first two terms and grouping the last two terms of a polynomial rewrite the expression as a product? Concept Check Factor by grouping. 1. a. 2x2 6x 5x 15 (x + 3)(2x + 5) b. 2x 5x 6x 15 (2x + 5)(x + 3) 2 2 2 2. a. 3 x 3xy xy y (x + y)(3x – y) 2 2 b. 3 x xy 3xy y (3x – y)(x + y) 3. a. 2a2 2ab 3ab 3b2 b. 2a 3ab 2ab 3b 2 2 (a – b)(2a – 3b) (2a – 3b)(a – b) 4. Compare your answers to parts (a) and (b) in Exercises 1–3 above. Do different groupings of the terms in a polynomial affect the binomial factoring? No Objective 5.5C Vocabulary to Review FOIL method [5.3B] trinomial [5.2A] New Vocabulary quadratic trinomial factoring a quadratic trinomial nonfactorable over the integers prime polynomial Concept Check Determine whether the statement is true or false. 1. The value of b in the trinomial x2 3x 5 is 3. False 2. To factor a trinomial of the form x2 bx c means to rewrite the polynomial as a product of two binomials. True 3. In factoring a trinomial, if the constant term of the trinomial is positive, then the signs of both binomial constants will be the same. True 4. In factoring a trinomial, if the constant term of the trinomial is negative, then the signs of both binomial constants will be negative. False 5. The first step in factoring a trinomial is to determine whether the terms of the trinomial have a common factor. True Optional Student Activity Complete the table by finding two integers whose product is given in the column headed ab and whose sum is given in the column headed a b. Assume a b . ab a+b 100 20 40 13 -42 -11 -72 -1 75 -20 44 -15 a b Column a: 10, 5, -14, -9, -15, -11 Column b: 10, 8, 3, 8, -5, -4 Objective 5.5D Vocabulary to Review quadratic trinomial [5.5C] factoring a quadratic trinomial [5.5C] Discuss the Concepts In factoring a trinomial of the form ax2 bx c by using trial factors, how are the signs of the last terms of the two binomial factors determined? Concept Check 1. The area of a rectangle is (2x2 + 9x + 9) in2. Find the dimensions of the rectangle in terms of the variable x. (2x + 3) in. by (x + 3) in. 2 2 2. The area of a rectangle is 3 x 16 x 5 mi Find the dimensions of the rectangle in terms of the variable x. (3x + 1) mi by (x + 5) mi 2 2 3. The area of a parallelogram is 30 x 21x 3 yd Find the dimensions of the parallelogram in terms of the variable x. (6x + 3) yd by (5x + 1) yd 2 2 4. The area of a parallelogram is 4 x 17 x 15 ft Find the dimensions of the parallelogram in terms of the variable x. (4x + 5) ft by (x + 3) ft Optional Student Activity 2 2 The area of a rectangle is 3x x 2 ft . Find the dimensions of the rectangle in terms of the variable x. Given that x > 0, specify the dimension that is the length and the dimension that is the width. Can x be less than 0? Can x be equal to 0? The dimensions are (3x – 2) ft by (x + 1) ft. If x = 1.5 then the rectangle is a square. If x < 1.5 the length is (x + 1) ft and the width is (3x – 2) ft. If x > 1.5 the width is (x + 1) ft and the length is (3x – 2) ft. If x < 0, then the area 3x2 + x – 2 is less than 0, which is not possible. Therefore, x cannot be less than 0. If x = 0, then the dimension 3x2 + x – 2 is negative, which is not possible. Therefore, x cannot be equal to 0. Objective 5.6A Vocabulary to Review term [1.3B] factor [5.5A] New Vocabulary perfect square square root of a perfect square difference of two perfect squares sum of two perfect squares product of the sum and difference of two terms perfect-square trinomial New Symbols Discuss the Concepts 1. Is x2 + 9 factorable? Why or why not? 2. Provide examples of the product of the sum and difference of two terms. For each example, state the two terms, the sum of the two terms, the difference of the two terms, and how the product is represented. 3. Is the product of the sum and difference of two terms always a binomial? 4. What is a perfect-square trinomial? 5. How can you determine the factors of a perfect-square trinomial? Concept Check 2 2 1. The area of a square is 4 x 12 x 9 cm . Find the length of a side of the square in terms of the variable x. (2x + 3) cm 2 2 2. The area of a square is 9 x 6 x 1 m . Find the length of a side of the square in terms of the variable (3x + 1) m x. Optional Student Activity 1. Find all integers k such that the trinomial is a perfect square. a. x2 kx 36 -12, 12 b. 4x kx 25 2 -20, 20 c. 49 x kxy 64 y 2 2 d. x2 8x k 16 e. x 12x k 2 f. x 4 xy ky 2 -112, 112 36 2 4 2 2 2. The area of a square is 16 x 24 x 9 ft . Find the dimensions of the square in terms of the variable x. Can x = 0? What are the possible values of x? (4x + 3) ft by (4x + 3) ft; Yes; x 3 4 Objective 5.6B New Vocabulary perfect cube cube root of a perfect cube sum of two perfect cubes difference of two perfect cubes New Symbols 3 Discuss the Concepts 1. Are both x2 16 and x3 27 factorable? Why or why not? 2. Which of the following are perfect cubes? Explain your answer. a. 125x8 12 b. 1y c. 8c9 d. 9b 27 3. How can you determine the factors of the sum of two perfect cubes? 4. How can you determine the factors of the difference of two perfect cubes? Optional Student Activity 1. What is the smallest positive integer by which 252 should be multiplied to obtain a perfect cube? 6 6 2. Find the quotient when x y is divided by x y. 3. Factor: ax3 b bx3 a (x – 1)(x2 + x + 1)(a + b) x5 + x4y + x3y2 + x2y3 + xy4 + y5 294 Objective 5.6C Vocabulary to Review quadratic trinomial [5.5C] New Vocabulary quadratic in form Concept Check Rewrite each of the following as the difference of two squares. Then factor. 1. 16 x2 1 (4x)2 – 12; (4x + 1)(4x – 1) 2. 9x4 25 (3x2)2 – 52; (3x2 + 5)(3x2 – 5) Rewrite each of the following in quadratic form. Then factor. 3. x4 3x2 2 4. 4x4 9x2 9 u2 + 3u + 2; (x2 + 2)(x2 + 1) 4u2 – 9u – 9; (4x2 + 3)(x2 – 3) Optional Student Activity 1. Factor: x4 64 4 2 Suggestion: Add and subtract 16x2 so that the expression becomes x 16 x 64 16x2 . Now factor by grouping. (x2 – 4x + 8)(x2 + 4x + 8) 4 2 2 4 2. Using the strategy in Exercise 1, factor x x y y . Suggestion: Add and subtract x 2 y 2 . (x2 + xy + y2)(x2 – xy + y2) Objective 5.6D Vocabulary to Review common factor [5.5A] binomial [5.2A] difference of two perfect squares [5.6A] sum of two perfect cubes [5.6B] difference of two perfect cubes [5.6B] trinomial [5.2A] perfect-square trinomial [5.6A] factoring by grouping [5.5B] nonfactorable over the integers [5.5C] prime polynomial [5.5C] Discuss the Concepts 1. Provide an example of each of the following: the difference of two perfect squares the product of the sum and difference of two terms a perfect-square trinomial the square of a binomial the sum of two perfect squares the sum of two perfect cubes the difference of two perfect cubes a prime polynomial 2. Can a third-degree polynomial have factors x 1, x 1, x 3, and x 4? Why or why not? Concept Check 2 1. The volume of a box is 2 xy 12 xy 10 x cubic inches. Find the dimensions of the box in terms of the variables x and y. 2x inches by (y + 1) inches by (y + 5) inches 2 2. The volume of a box is 3 x y 21xy 36 y the variables x and y. cubic centimeters. Find the dimensions of the box in terms of 3y centimeters by (x + 4) centimeters by (x + 3) centimeters Optional Student Activity 1. Find the least common multiple of the polynomials 3x 2 x 2, 3x 2 8 x 4, and x3 2 x2 x 2. 3x4 – 8x3 + x2 + 8x – 4 2. Factor: x2 x 4 2 x2 5x 12 (x – 4)(x – 3)(x + 1) Answers to Writing Exercises 131. If x 3 and x 4 are factors of x3 6x2 7 x 60 , then x3 6x2 7 x 60 is divisible by x 3 and x 4. Divide x3 6x 2 7 x 60 by x 3. The quotient is x2 9x 20. Divide this quotient by x 4. The quotient is x 5. Therefore, x 5 is a third first-degree factor of x3 6x2 7x 60. Objective 5.7A Vocabulary to Review descending order [5.2A] New Vocabulary quadratic equation quadratic equation in standard form Properties to Review Multiplication Property of Zero [1.3A] New Rules Principle of Zero Products Concept Check 1. Solve for the largest positive root of the equation 2x3 x2 8x 4. 2. Show that the solutions of the equation ax2 bx 0 are 0 and 2 b . a Factor the left side: x(ax + b) = 0; then x = 0 or ax + b = 0. Solve ax + b = 0 for x: x Optional Student Activity Solve for x. 1. x2 9ax 14a2 0 2a, 7a b a 2. x 2 9 xy 36 y 2 0 -12y, 3y 3. 3x2 4cx c2 0 c ,c 3 4. 2x2 3bx b2 0 b , -b 2 Objective 5.7B Optional Student Activity 1. Find two consecutive integers whose cubes differ by 127. 6 and 7 or -7 and -6 2. The sum of the squares of three consecutive odd integers is 83. Find the three integers. 3, -5, and -7 3, 5, and 7 or - 3. A model for the height above the ground of an arrow projected into the air with an initial velocity of 120 ft/s is h = -16t2 + 120t + 5, where h is the height, in feet, of the arrow t seconds after it is released from the bow. Determine at what times the arrow is 181 ft above the ground. After 2 s and after 5.5 s 2 4. The base of a triangle is 2 in. less than four times the height. The area of the triangle is 45 in . Find the height and the length of the base of the triangle. Height: 5 in.; base: 18 in. 2 5. The height above Earth of a projectile fired upward is given by the formula s vo t 16t , where s is the height in feet, v0 is the initial velocity, and t is the time in seconds. Find the time for a projectile to return to Earth if it has an initial velocity of 200 ft/s. 12.5 s Answers to Writing Exercises 46. The error is in the division in Step 4. Because a b 0, dividing by a b is equivalent to dividing by 0, but division by zero is undefined. Answers to Focus on Problem Solving: Find a Counterexample 1. False. 0 is a real number, and 02 0 is not positive. 2. True 3. True 4. False. Let x = -4 and y 2. Then the expression (-4)2 < 22 is not true. 5. False. It is impossible to construct a triangle with a 2 , b 3, and c 10. In a triangle, the sum of the lengths of two sides must be greater than the length of the third side. 6. False. The product 3 3 3. 7. False. For n = 4, 1 ∙ 2 ∙ 3 ∙ 4 + 1 = 25, which is not a prime number. 8. False. Part of line segment AB lies outside the polygon. A B 9. True 10. False. If the points are selected such that AC + CD < DB, a triangle cannot be formed. Answers to Projects and Group Activities: Astronomical Distances and Scientific Notation 1. 5.865696 1012 mi 2. 4.26 light-years 3. 1.64239488 1021 mi 4. 1.96 104 A.U. 5. 63,000 A.U. 6. Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune, Pluto 7. Jupiter, Saturn, Neptune, Uranus, Earth, Venus, Mars, Mercury, Pluto 8. When we compare two numbers written in scientific notation, the number with the higher exponent on 10 is the larger number. If the exponents on 10 are the same, compare the numbers between 1 and 10; the number with a larger number between 1 and 10 is the larger number.
