Polynomials. Polynomials

Transcription

1 Preview of Algebra 1 Polynomials 1A Introduction to Polynomials 1-1 Polynomials LAB Model Polynomials 1- Simplifying Polynomials 1B Polynomial Operations LAB Model Polynomial Addition 1-3 Adding Polynomials LAB Model Polynomial Subtraction 1- Subtracting Polynomials 1-5 Multiplying Polynomials by Monomials LAB Multiply Binomials 1-6 Multiplying Binomials KEYWORD: MT8CA Ch1 Polynomials can be used to calculate the height of fireworks. Walt Disney Concert Hall, Los Angeles 586 Chapter 1

2 Associative Vocabulary Property Choose the best term from the list to complete each sentence. coefficient 1.? have the same variables raised to the same powers. Distributive. In the epression, is the?. Property 3. 5 ( 3) (5 ) 3 by the?. like terms ( ) by the?. Commutative Property Complete these eercises to review skills you will need for this chapter. Integer Operations Add or subtract () (5) (5) (19) (5) Evaluate Epressions Evaluate the epression for the given value of the variable. 1. ( y ) z for 8, y 3, z 15. 7ab 5 for a 3, b 16. (n 3) for n (t ) for t Simplify Algebraic Epressions Simplify each algebraic epression b a 11 3a 1 0. n 10m 9n Area of Squares, Rectangles, and Triangles Find the area of each figure cm 6 in. 36 cm m 15 in. m Polynomials 587

3 The information below unpacks the standards. The Academic Vocabulary is highlighted and defined to help you understand the language of the standards. Refer to the lessons listed after each standard for help with the math terms and phrases. The Chapter Concept shows how the standard is applied in this chapter. California Standard Preview of Algebra Students add, subtract, multiply, and divide monomials and polynomials. Students solve multistep problems, including word problems, by using these techniques. (Lessons 1-3, 1-, 1-5, 1-6; Labs 1-3, 1-, 1-6) Academic Vocabulary multistep more than one step technique a way of doing something Chapter Concept You use your knowledge of eponents to add, subtract, and multiply polynomials, and you use polynomials to solve problems. Eample: You simplify epressions such as and (3 5 3 ). Standards AF1. and AF1.3 are also covered in this chapter. To see these standards unpacked, go to Chapter 1, p. (AF1.) and Chapter 3, p. 11 (AF1.3). 588 Chapter 1

4 Study Strategy: Study for a Final Eam A cumulative final eam will cover material you have learned over the course of the year. You must be prepared if you want to be successful. It may help you to make a study timeline like the one below. weeks before the final: Look at previous eams and homework to determine areas I need to focus on; rework problems that were incorrect or incomplete. Make a list of all formulas I need to know for the final. Create a practice eam using problems from the book that are similar to problems from each eam. 1 week before the final: Take the practice eam and check it. For each problem I miss, find two or three similar problems and work those. Work with a friend in the class to quiz each other on formulas from my list. 1 day before the final: Make sure I have pencils and scratch paper. Try This Complete the following to help you prepare for your cumulative test. 1. Create a timeline that you will use to study for your final eam. Polynomials 589

5 1-1 Polynomials California Standards Preview of Algebra 1 Preparation for 10.0 Students add, subtract, multiply, and divide monomials and polynomials. Students solve multistep problems, including word problems, by using these techniques. Also covered: 7AF1. Why learn this? You can use polynomials to find the height of a firework when it eplodes. (See Eample.) Recall that a monomial is a number, variable, or a product of numbers and variables with eponents that are whole numbers. Monomials n, 3, a b 3, 7 Not monomials p.,,, g 5 EXAMPLE 1 Vocabulary polynomial bionomial trinomial degree of a polynomial Identifying Monomials Determine whether each epression is a monomial. 1 3 y 7 10y 0.3 monomial not a monomial and 7 are whole numbers. 0.3 is not a whole number. A polynomial is one monomial or the sum or difference of monomials. A simplified polynomial can be classified by the number of monomials, or terms, that it contains. A monomial has 1 term, a binomial has terms, and a trinomial has 3 terms. EXAMPLE Classifying Polynomials by the Number of Terms Classify each epression as a monomial, a binomial, a trinomial, or not a polynomial h 19.55g binomial Polynomial with terms 3 y monomial Polynomial with 1 term 6 y can be written as 1, where the not a polynomial eponent is not a whole number. 7mn m 5n trinomial Polynomial with 3 terms 590 Chapter 1 Polynomials

6 The degree of a term is the sum of the eponents of the variables in the term. The degree of a polynomial is the same as the degree of the term with the greatest degree. A polynomial can be classified by its degree. 5 y { { Degree Degree 5 Degree Degree 0 13 Degree 5 EXAMPLE 3 Classifying Polynomials by Their Degrees Find the degree of each polynomial Degree Degree 1 Degree 0 The greatest degree is, so the degree of 6 3 is. 6 3m m 5 6 3m m 5 Degree 0 Degree Degree 5 The greatest degree is 5, so the degree of 6 3m m 5 is 5. To evaluate a polynomial, substitute the given number for each variable. EXAMPLE A polynomial is an algebraic epression. For help with evaluating algebraic epressions, see Lesson 1-1. Physics Application The height in feet of a firework launched straight up into the air from s feet off the ground at velocity v after t seconds is given by the polynomial 16t vt s. Find the height of a firework launched from a 10 ft platform at 00 ft/s after 5 seconds. 16t vt s Write the polynomial epression for height. 16(5) 00(5) 10 Substitute 5 for t, 00 for v, and 10 for s Simplify. The firework is 610 feet high 5 seconds after launching. Think and Discuss 1. Describe two ways you can classify a polynomial. Give a polynomial with three terms, and classify it two ways.. Eplain why 5 3 is a polynomial but 5 3 is not. 1-1 Polynomials 591

7 1-1 Eercises See Eample 1 GUIDED PRACTICE California Standards Practice Preparation for Algebra ; 7AF1. Determine whether each epression is a monomial. 1. y KEYWORD: MT8CA 1-1 KEYWORD: MT8CA Parent See Eample See Eample 3 See Eample Classify each epression as a monomial, a binomial, a trinomial, or not a polynomial y 6. 5r 3r Find the degree of each polynomial. 9. 7m 5 3m The trinomial 16t t 7 describes the height in feet of a ball thrown straight up from a 7 ft platform with a velocity of ft/s after t seconds. What is the ball s height after seconds? See Eample 1 See Eample See Eample 3 See Eample INDEPENDENT PRACTICE Determine whether each epression is a monomial y y Classify each epression as a monomial, a binomial, a trinomial, or not a polynomial m n g 3 1 h a v 3 s Find the degree of each polynomial m m p 7p 9. n 30. 3y The volume of a bo with height, length, and width 3 5 is given by the trinomial What is the volume of the bo if its height is inches? Etra Practice See page EP. PRACTICE AND PROBLEM SOLVING 3. Transportation The distance in feet required for a car traveling at r r mi/h to come to a stop can be approimated by the binomial r. 0 About how many feet will be required for a car to stop if it is traveling at 70 mi/h? 59 Chapter 1 Polynomials

8 Classify each epression as a monomial, a binomial, a trinomial, or not a polynomial. If it is a polynomial, give its degree y 6y 37. f 3 5f 5 f b 9b 8b Transportation Gas mileage at speed s in miles per hour can be estimated using the given polynomials. Evaluate the polynomials to complete the table. Compact Midsize Van 0.05s +.5s s + 1.5s s +.9s 53 Gas Mileage (mi/gal) 0 mi/h 50 mi/h 60 mi/h 6. Reasoning Without evaluating, tell which of the following binomials has the greatest value when 10. Eplain what method you used. A B C 3 8 D What s the Error? A student says that the degree of the polynomial b 5 7b 9 6b is 5. What is the error? 8. Write About It Give some eamples of words that start with mono-, bi-, tri-, and poly-, and relate the meaning of each to polynomials. 9. Challenge The base of a triangle is described by the binomial, and its height is described by the trinomial 3 7. What is the area of the triangle if 5? NS1.1, NS., AF Multiple Choice The height in feet of a soccer ball kicked straight up into the air from s feet off the ground at velocity v after t seconds is given by the trinomial 16t vt s. What is the height of the soccer ball kicked from feet off the ground at 90 ft/s after 3 seconds? A 3 ft B 15 ft C 90 ft D 18 ft 51. Gridded Response What is the degree of the polynomial 6 7k 8k 9? Write each number in scientific notation. (Lesson -5) 5.,080, ,910,000,000 Find the two square roots of each number. (Lesson -6) Polynomials 593

9 Model Polynomials 1-1 Use with Lesson 1-1 KEY REMEMBER KEYWORD: MT8CA Lab1 0 California Standards Preview of Algebra 1 Preparation for 10.0 Students add, subtract, multiply, and divide monomials and polynomials. Students solve multistep problems, including word problems, by using these techniques. You can use algebra tiles to model polynomials. To model the polynomial 3, you need four -tiles, one -tile, and three 1-tiles Activity 1 1 Use algebra tiles to model the polynomial 6. All signs are positive, so use all yellow tiles Chapter 1 Polynomials

10 Use algebra tiles to model the polynomial 6. Modeling 6 is similar to modeling 6. Remember to use red tiles for negative values Think and Discuss 1. How do you know when to use red tiles? Try This Use algebra tiles to model each polynomial Activity 1 Write the polynomial modeled by the algebra tiles below The polynomial modeled by the tiles is Think and Discuss 1. How do you know the coefficient of the -term in Activity? Try This Write a polynomial modeled by each group of algebra tiles Hands-On Lab 595

11 1- Simplifying Polynomials California Standards Preview of Algebra 1 Preparation for 10.0 Students add, subtract, multiply, and divide monomials and polynomials. Students solve multistep problems, including word problems, by using these techniques. Also covered: 7AF1.3 Why learn this? You can simplify polynomials to determine the amount of lumber that can be harvested from a tree. (See Eample.) You can simplify a polynomial by adding or subtracting like terms. Like terms The variables have the same powers. Not like terms The variables have different powers. EXAMPLE 1 Identifying Like Terms Identify the like terms in each polynomial. a a 3 5a 6a a a 3 5a 6a Like terms: a and 5a; a and 6a 5 y y y 3 5 y y y 3 Like terms: 5 y 3, 1 5 y 3, and 6 5 y 3 5m 3mn m 5m 3mn m There are no like terms. Identify like terms. Identify like terms. Identify like terms. To simplify a polynomial, combine like terms. First arrange the terms from highest degree to lowest degree by using the Commutative Property. EXAMPLE When you rearrange terms, move the operation symbol in front of each term with that term. Simplifying Polynomials by Combining Like Terms Simplify Identify like terms. Commutative Property Combine coefficients: and Chapter 1 Polynomials

12 Simplify. 5a b 1ab a b ab 3ab 5a b 1ab a b ab 3ab Identify like terms. 5a b a b 1ab ab 3ab 9a b 11ab 3ab Commutative Property Combine coefficients: 5 9 and You may need to use the Distributive Property to simplify a polynomial. EXAMPLE 3 Simplifying Polynomials by Using the Distributive Property Simplify. (3 5) (3 5) Distributive Property (3 ) (5) 1 0 No like terms (ab 5b) 3ab 6 (ab 5b) 3ab 6 (ab ) (5b) 3ab 6 8ab 10b 3ab 6 11ab 10b 6 Distributive Property Identify like terms. Combine coefficients. EXAMPLE Business Application A board foot is equal to the volume of a 1 ft by 1 ft by 1 in. piece of lumber. The amount of lumber that can be harvested from a tree with diameter d in. is approimately (d 3 30d 300d 1000) board feet. Use the Distributive Property to write an equivalent epression (d 3 30d 300d 1000) d d 1.5d d d 1.5d Think and Discuss 1. Tell how you know when you can combine like terms.. Give an eample of an epression that you could simplify by using the Distributive Property. Then give an epression that you could simplify by combining like terms. 1- Simplifying Polynomials 597

13 1- Eercises California Standards Practice Preparation for Algebra ; 7AF1.3 KEYWORD: MT8CA 1- KEYWORD: MT8CA Parent GUIDED PRACTICE See Eample 1 Identify the like terms in each polynomial. 1. 3b 5b b b 6. 7mn 5m n 8m n m n See Eample Simplify b b 7b 9 b 3b See Eample 3 See Eample 5. (3 8) 6. 7( ) 7. 5(3a 5a) a a 8. The level of nitric oide emissions, in parts per million, from a car engine is approimated by the polynomial 0,000 5(800 ), where is the air-fuel ratio. Use the Distributive Property to write an equivalent epression. INDEPENDENT PRACTICE See Eample 1 Identify the like terms in each polynomial. 9. t t 5t 5t 10. 8rs 3r s 5r s rs 5 See Eample Simplify. 11. p 3p 5p 1p 1. 3fg f g fg 3fg f g 6fg See Eample 3 See Eample Etra Practice See page EP ( 5) 7 1. (b 3) 5b 3b (6y3 8) 3y The concentration of a certain medication in an average person s bloodstream h hours after injection can be estimated using the epression 6(0.03h 0.00h 0.01h 3 ). Use the Distributive Property to write an equivalent epression. PRACTICE AND PROBLEM SOLVING Simplify. 17. s 3s 10s 5s gh g h g h g h 19. ( 5 ) ( 5 3 ) 3 1. (m 3m ) 7(3m m). 6b b 3(b 6) 3. 5mn 3m 3 n 3(m 3 n mn). 3( y) (3 y) 598 Chapter 1 Polynomials 5. Life Science The rate of flow in cm/s of blood in an artery at d cm from the center is given by the polynomial 1000(0.0 d ). Use the Distributive Property to write an equivalent epression.

14 Art Abstract artists often use geometric shapes, such as cubes, prisms, pyramids, and spheres, to create sculptures. 6. Suppose the volume of a sculpture is approimately s 3 0.5s s s 3 cm 3 and the surface area is approimately 6s 3.1s 7.6s 3.s cm. a. Simplify the polynomial epression for the volume of the sculpture, and find the volume of the sculpture for s 5. b. Simplify the polynomial epression for the surface area of the sculpture, and find the surface area of the sculpture for s 5. Balanced/Unbalanced O by Fletcher Benton 7. A sculpture features a large ring with an outer lateral surface area of about y in, an inner lateral surface area of about 38y in, and bases, each with an area of about 1y in. Write and simplify a polynomial that epresses the surface area of the ring. 8. Challenge The volume of the ring on the sculpture from Eercise 7 is 9πy 36πy in 3. Simplify the polynomial, and find the volume for 1 and y 7.5. Give your answer both in terms of π and to the nearest tenth. Pyramid Balancing Cube and Sphere, artist unknown KEYWORD: MT8CA Art NS1.3, AF Multiple Choice Simplify the epression A B C D Short Response Identify the like terms in the polynomial 3 5. Then simplify the polynomial. Find each percent to the nearest tenth. (Lesson 6-3) 31. What percent of 8 is? 3. What percent of 195 is 16? Create a table for each quadratic function, and use it to graph the function. (Lesson 7- ) 33. y 1 3. y 1 1- Simplifying Polynomials 599

15 Quiz for Lessons 1-1 Through Polynomials Determine whether each epression is a monomial c d 8 Classify each epression as a monomial, a binomial, a trinomial, or not a polynomial a 3 a y Find the degree of each polynomial. 7. u c c 5 c The depth, in feet below the ocean surface, of a submerging eploration submarine after y minutes can be approimated by the polynomial 0.001y 0.1y 3 3.6y. Estimate the depth after 5 minutes. 1- Simplifying Polynomials Identify the like terms in each polynomial y y y 1. z 7z z z t 8 t ab 3ac 5bc ac 6ab Simplify b 5 b 3 7 5b y y 7y y ( 7) y 5 5y (5y ) 19. The area of one face of a cube is given by the epression 3s 5s. Write a polynomial to represent the total surface area of the cube. 0. The area of each lateral face of a regular square pyramid is given by the epression 1 b b. Write a polynomial to represent the lateral surface area of the pyramid. 600 Chapter 1 Polynomials

16 California Standards MR.1 Use estimation to verify the reasonableness of calculated results. Also covered: NS1.7 Look Back Estimate to check that your answer is reasonable Before you solve a word problem, you can often read through the problem and make an estimate of the correct answer. Make sure your answer is reasonable for the situation in the problem. After you have solved the problem, compare your answer with the original estimate. If your answer is not close to your estimate, check your work again. Each problem below has an incorrect answer given. Eplain why the answer is not reasonable, and give your own estimate of the correct answer. 1 The perimeter of rectangle ABCD is 8 cm. What is the value of? A 9 B C Answer: 5 D A patio layer can use 6y ft of accent edging to divide a patio into three sections measuring ft long by y ft wide. If each section must be at least 15 ft long and have an area of at least 165 ft, what is the minimum amount of edging needed for the patio? 3 y ft ft A baseball is thrown straight up from a height of 3 ft at 30 mi/h. The height of the baseball in feet after t seconds is 16t t 3. How long will it take the baseball to reach its maimum height? Answer: 5 minutes Jacob deposited $000 in a savings account that earns 6% simple interest. How much money will he have in the account after 7 years? Answer: $195 Answer: 5 ft Focus on Problem Solving 601

17 1-3 Model Polynomial Addition Use with Lesson 1-3 KEY REMEMBER KEYWORD: MT8CA Lab1 0 California Standards Preview of Algebra Students add, subtract, multiply, and divide monomials and polynomials. Students solve multistep problems, including word problems, by using these techniques. You can use algebra tiles to model polynomial addition. Activity 1 Use algebra tiles to find ( 3) ( 5). 3 5 Use tiles to represent all terms from both epressions. Remove any zero pairs. The remaining tiles represent the sum 3. 3 Think and Discuss 1. Eplain what happens when you add the -terms in ( 5) ( ). Try This Use algebra tiles to find each sum. 1. (3m m 6) (m m 3). (5b b 1) (b 1) 60 Chapter 1 Polynomials

18 1-3 Adding Polynomials California Standards Preview of Algebra Students add, subtract, multiply, and divide monomials and polynomials. Students solve multistep problems, including word problems, by using these techniques. Also covered: 7AF1.3 Why learn this? You can add polynomials to find the amount of material needed to mat and frame a picture. (See Eample 3.) Adding polynomials is similar to simplifying polynomials. One way to add polynomials is to write them horizontally. First write the polynomials as one polynomial, and then use the Commutative Property to combine like terms. EXAMPLE 1 Adding Polynomials Horizontally Add. (6 3 ) (7 6) (cd 3cd 6) (7cd 6cd 6) cd 3cd 6 7cd 6cd 6 cd 6cd 3cd 7cd cd cd (ab a) (3ab a 3) (a 5) ab a 3ab a 3 a 5 ab 3ab a a a 3 5 ab 9a Write as one polynomial. Commutative Property Combine like terms. Write as one polynomial. Commutative Property Combine like terms. Write as one polynomial. Commutative Property Combine like terms. You can also add polynomials in a vertical format. Write the second polynomial below the first one. Be sure to line up the like terms. If the terms are rearranged, remember to keep the correct sign with each term. 1-3 Adding Polynomials 603

19 EXAMPLE Adding Polynomials Vertically Add. (5a a ) (a 3a 1) 5a a a 3a 1 Place like terms in columns. 9a 7a 3 Combine like terms. (y 3 y) (8y 3) y 3 y 8y 3 Place like terms in columns. 10y y 3 Combine like terms. (a b 3a 6ab) (ab a 5) (3 7ab) a b 3a 6ab a ab 5 7ab 3 Place like terms in columns. a b a 3ab Combine like terms. EXAMPLE 3 Reasoning Art Application 1 in. Mina is putting a mat of width m and a frame of width f around f 11 in. an 11-inch by 1-inch picture. Find m m f an epression for the amount of framing material she needs. m The amount of material Mina f needs equals the perimeter of the outside of the frame. Draw a diagram to help you determine the outer dimensions of the frame. Width 1 m m f f 1 m f P (11 m f ) (1 m f ) m f 8 m f 50 8m 8f Length 11 m m f f 11 m f P w She will need 50 8m 8f inches of framing material. f m Simplify. Combine like terms. Think and Discuss 1. Compare adding (5 ) (3 ) vertically with adding it horizontally.. Eplain how adding polynomials is similar to simplifying polynomials. 60 Chapter 1 Polynomials

20 1-3 Eercises See Eample 1 GUIDED PRACTICE Add. 1. ( ) (3 7) California Standards Practice Preview of Algebra ; 7AF1.3 KEYWORD: MT8CA 1-3 KEYWORD: MT8CA Parent. ( 6) (1 3) 3. (r s 3rs) (r s 8rs) (6r s 1rs) See Eample. (b 5b 10) (6b 7b 8) 5. (9ab 5ab 6a b) (8ab 1a b 6) (6ab 5a b 1) 6. (h j hj 3 hj 6) (5hj 3 5) (6h j 7hj) See Eample 3 7. Colette is putting a mat of width 3w and a frame of width w around a 16-inch by 8-inch poster. Find an epression for the amount of frame material she needs. 16 in. 3w 8 in. w w 3w See Eample 1 INDEPENDENT PRACTICE Add. 8. (5 y y 3) (7y 3 y) 9. (5g 9) (7g g 8) 10. (6bc b c 8bc ) (6bc 3bc ) 11. (9h 5h h 6 ) (h 6 6h 3h ) 1. (pq 5p q 9pq ) (6p q 11pq ) (pq 7pq 6p q) See Eample 13. (8t t 3) (5t 8t 9) 1. (5b 3 c 3b c bc) (8b 3 c 3bc 1) (b c 5bc 9) 15. (w 3w 5) (w 3w 1) (w w 6) See Eample Each side of an equilateral triangle has length w 3. Each side of a square has length w. Write an epression for the sum of the perimeter of the equilateral triangle and the perimeter of the square. w 3 w 1-3 Adding Polynomials 605

21 Etra Practice See page EP5. PRACTICE AND PROBLEM SOLVING Add. 17. (3w y 3wy wy) (5wy wy 7w y) (wy 5wy 3w y) 18. (p t 3pt 5) (p t pt 3pt) (1 5pt p t) Business 19. Geometry Write and simplify an epression for the combined volumes of a sphere with volume 3 πr 3, a cube with volume r 3, and a prism with volume r 3 r 5r. Use 3.1 for π. 0. Business The cost of producing n toys at a factory is given by the polynomial 0.5n 3n 1. The cost of packaging is 0.5n 5n. Write and simplify an epression for the total cost of producing and packaging n toys. According to the Toy Industry Association, $.6 billion was spent on toys worldwide in 000. KEYWORD: MT8CA Toys 1. Reasoning Two airplanes depart from the same airport, traveling in opposite directions. After hours, one airplane is 00 miles from the airport, and the other airplane is miles from the airport. How could you determine the distance between the two planes? Eplain.. Write two polynomials whose sum is 3m m Choose a Strategy What is the missing term? (6 3) (3 5) A B C 10 D 10. Write a Problem A plane leaves an airport heading north at 3 mi/h. At the same time, another plane leaves the same airport, heading south at mi/h. Write a problem using the speeds of both planes. 5. Write About It Eplain how to add polynomials. 6. Challenge What polynomial would have to be added to 6 5 so that the sum is 3 7? NS1.5, MG1. 7. Multiple Choice Debbie is putting a deck of width 5w around her 0 foot by 80 foot pool. Which is the epression for the perimeter of the pool and deck combined? A w w 00 0w 50 5w B C D 8. Gridded Response What is the sum of ( ), ( ), and ( )? Find the fraction equivalent of each decimal. (Lesson -1) Using the scale 1 in. 6 ft, find the height or length of each object. (Lesson 5-7) 33. a 1 in. tall model of an office building 3. a.5 in. long model of a train 606 Chapter 1 Polynomials

22 1- Use with Lesson 1- Model Polynomial Subtraction KEY REMEMBER KEYWORD: MT8CA Lab1 0 California Standards Preview of Algebra Students add, subtract, multiply, and divide monomials and polynomials. Students solve multistep problems, including word problems, by using these techniques. You can use algebra tiles to model polynomial subtraction. Activity 1 Use algebra tiles to find ( 3) ( 3). 3 3 Remember, subtracting is the same as adding the opposite. Use the opposite of each term in Remove any zero pairs. The remaining tiles represent the difference 3 6. Think and Discuss 1. Why do you have to add the opposite when subtracting? Try This Use algebra tiles to find each difference. 1. (6m m) (m ). (5b 9) (b 9) 1- Hands-On Lab 607

23 1- Subtracting Polynomials California Standards Preview of Algebra Students add, subtract, multiply, and divide monomials and polynomials. Students solve multistep problems, including word problems, by using these techniques. Also covered: 7AF1.3 Who uses this? Manufacturers can subtract polynomials to estimate the cost of making a product and the revenue from sales. (See Eample.) Recall that to subtract an integer, you add its opposite. To subtract a polynomial, you first need to find its opposite. EXAMPLE 1 Finding the Opposite of a Polynomial Find the opposite of each polynomial. 8 3 y 6 z (8 3 y 6 z) The opposite of a positive term 8 3 y 6 z is a negative term. 1 5 (1 5) 1 5 3ab ab 3 (3ab ab 3) 3ab ab 3 Distributive Property Distributive Property To subtract a polynomial, add its opposite. EXAMPLE Subtracting Polynomials Horizontally Subtract. (n 3 n 5n ) (7n n 9) (n 3 n 5n ) (7n n 9) Add the opposite. n 3 5n n n 7n 9 Commutative Property n 3 9n 8n 9 Combine like terms. (cd cd ) (7cd 5cd ) (cd cd ) (7cd 5cd) cd 7cd cd 5cd 5cd 6cd Add the opposite. Commutative Property Combine like terms. 608 Chapter 1 Polynomials

24 You can also subtract polynomials in a vertical format. Write the second polynomial below the first one, lining up the like terms. EXAMPLE 3 Subtracting Polynomials Vertically Subtract. ( 3 1) ( ) ( 3 1) 3 1 ( ) Add the opposite. 5 3 (m n 3mn m) (8m n 6mn 3) (m n 3mn m) m n 3mn m (8m n 6mn 3) 8m n 6mn 3 Add the 1m n 3mn m 3 opposite. ( y y 6) (7 5y 6) ( y y 6) y y 6 (7 5y 6) 5y 7 6 Rearrange terms y y 13 6 as needed. EXAMPLE Business Application Suppose the cost in dollars of producing model kits is given by the polynomial 3 00,000 and the revenue generated from sales is given by the polynomial Find a polynomial epression for the profit from making and selling model kits, and evaluate the epression for 00, (3 00,000) (3 00,00) , ,000 revenue cost Add the opposite. Commutative Property Combine like terms. The profit is given by the polynomial ,000. For 00,000, (00,000) 17(00,000) 00,000 1,00,000. The profit is $1,00,000, or $1. million. Think and Discuss 1. Eplain how to find the opposite of a polynomial.. Compare subtracting polynomials with adding polynomials. 1- Subtracting Polynomials 609

25 1- Eercises California Standards Practice Preview of Algebra ; 7AF1.3 KEYWORD: MT8CA 1- KEYWORD: MT8CA Parent GUIDED PRACTICE See Eample 1 Find the opposite of each polynomial. 1. y. 5 y y y y y See Eample Subtract. 7. (b 3 5b 8) (b 3 b 1) 8. 7b (b 3b 1) 9. (m n 7mn 3mn ) (5mn m n) See Eample (8 1) (5 3) 11. ( y y 3 ) (y 7 ) 1. (5ab ab 3a b) (7 5ab 3ab a b) See Eample 13. The volume of a rectangular prism, in cubic inches, is given by the epression The volume of a smaller rectangular prism is given by the epression How much greater is the volume of the larger rectangular prism? Evaluate the epression for 3. See Eample 1 INDEPENDENT PRACTICE Find the opposite of each polynomial. 1. rn 15. 3v 5v 16. m 6m 17. y y 18. 8n 6 5n 3 n 19. 9b b 9 See Eample Subtract. 0. (6w 3w 6) (3w w 5) 1. (1a a ) (8 a 9a). (7r s 5rs 6r s 7rs) (3rs 3r s 8rs) See Eample 3 3. ( 6 1) (3 9 5). (3a b ab a ) (a b 5a 3b 6) 5. (pt 6p 3 5p t ) (5p 6pt 7p t ) See Eample 6. The current in an electrical circuit at t seconds is t 3 5t t 00 amperes. The current in another electrical circuit is 3t 3 t 5t 100 amperes. Write an epression to show the difference in the two currents. 610 Chapter 1 Polynomials

26 Etra Practice See page EP5. PRACTICE AND PROBLEM SOLVING Subtract. 7. (6a 3b 5ab) (6a 5b 7ab) 8. (pq 6p q 3pq) (7pq 7p q 3pq) 9. (9y 5 y ) (3y 7 y ) 30. The area of the rectangle is a a 5 cm. The area of the square is a a 6 cm. What is the area of the shaded region? 31. The area of the square is 6 in. The area of the triangle is 5 in. What is the area of the shaded region? 3. Business The price in dollars of one share of stock after y years is modeled by the epression 3y 3 6y.5. The price of one share of another stock is modeled by 3y 3 y 5.5. What epression shows the difference in price of the two stocks after y years? 33. Choose a Strategy Which polynomial has the greatest value when 6? A B 3 8 C D Write About It Eplain how to subtract the polynomial from Challenge Find the values of a, b, c, and d that make the equation true. (t 3 at bt 6) (ct 3 t 7t 1) t 3 5t 15t d AF1., AF1.3, AF. 36. Multiple Choice What is the opposite of the polynomial a b 3ab 5ab? A a b 3ab 5ab C a b 3ab 5ab B a b 3ab 5ab D a b 3ab 5ab 37. Etended Response A square has an area of A triangle inside the square has an area of. Create an epression for the area of the square minus the area of the triangle. Evaluate the epression for 8. Multiply. (Lesson -) 38. (3)(6) 39. (9m 3 )(7m ) 0. (8ab )(5a ) 1. (r s)(r 6 s 9 ) Simplify. (Lesson 1-). 3 y y 3 y 3. (zy 3 zy) 3zy 5zy 3. 6(3 6 1) 1- Subtracting Polynomials 611

27 1-5 Multiplying Polynomials by Monomials California Standards Preview of Algebra Students add, subtract, multiply, and divide monomials and polynomials. Students solve multistep problems, including word problems, by using these techniques. Also covered: 7AF1., 7AF1.3, 7AF. Why learn this? You can multiply polynomials and monomials to determine the dimensions of a planter bo. (See Eample 3.) Remember that when you multiply two powers with the same bases, you add the eponents. To multiply two monomials, multiply the coefficients and add the eponents of the variables that are the same. (5m n 3 )(6m 3 n 6 ) 5 6 m 3 n m 5 n 9 EXAMPLE 1 Multiplying Monomials Multiply. (r 3 s )(6r 5 s 6 ) 6 r 3 5 s 6 Multiply coefficients. Add eponents that r 8 s 10 have the same base. (9 y)( 3 yz 6 ) 9 3 y 1 1 z 6 Multiply coefficients. Add eponents that 18 5 y z 6 have the same base. To multiply a polynomial by a monomial, use the Distributive Property. Multiply every term of the polynomial by the monomial. EXAMPLE Multiplying a Polynomial by a Monomial Multiply. 1 (y z) 1 (y z) Multiply each term in the parentheses by 1. 1 y 1 z When multiplying a polynomial by a negative monomial, be sure to distribute the negative sign. 5a b(3a b 3 6a b 3 ) 5a b(3a b 3 6a b 3 ) Multiply each term in the parentheses 15a 6 b 30a b by 5a b. 61 Chapter 1 Polynomials

28 Multiply. 5rs (r s 3rs 3 rst) 5rs (r s 3rs 3 rst) Multiply each term in the 5r 3 s 6 15r s 5 0r s 3 t parentheses by 5rs. EXAMPLE 3 PROBLEM SOLVING APPLICATION Chrystelle is making a planter bo with a square base. She wants the height of the bo to be 3 inches more than the side length of the base. If she wants the volume of the bo to be 680 in 3, what should the side length of the base be? Reasoning 1. 1 Understand the Problem If the side length of the base is s, then the height is s 3. The volume is s s (s 3) s (s 3). The answer will be a value of s that makes the volume of the bo equal to 680 in Make a Plan You can make a table of values for the polynomial to try to find the value of s. Use the Distributive Property to write the epression s (s 3) another way. Use substitution to complete the table Solve s (s 3) s 3 3s Distributive Property s s 3 3s (15) (16) (17) (18) The side length of the base should be 18 inches. 1. Look Back If the side length of the base were 18 inches and the height were 3 inches more, or 1 inches, then the volume would be in 3. The answer is reasonable. Think and Discuss 1. Compare multiplying two monomials with multiplying a polynomial by a monomial. 1-5 Multiplying Polynomials by Monomials 613

29 1-5 Eercises See Eample 1 See Eample GUIDED PRACTICE California Standards Practice Preview of Algebra ; 7AF1., 7AF1.3, 7AF. Multiply. 1. (5s t )(3st 3 ). ( y 3 )(6 y 3 ) 3. (5h j )(7h j 6 ). 6m(m 5 ) 5. 7p 3 r(5pr ) 6. 13g 5 h 3 (10g 5 h ) 7. h(3m h) 8. ab(a b ab ) KEYWORD: MT8CA 1-5 KEYWORD: MT8CA Parent 9. 3( 5 10) 10. 6c d(3cd 3 5c 3 d cd ) See Eample 3 See Eample The formula for the area of a trapezoid is A 1 h(b 1 b ), where h is the trapezoid s height and b 1 and b are the lengths of its bases. Use the Distributive Property to simplify the epression. Then use the epression to find the area of a trapezoid with height 1 in. and base lengths 9 in. and 7 in. INDEPENDENT PRACTICE Multiply. 1. (6 y 5 )(3y ) 13. (gh 3 )(g h 5 ) 1. (a b)(b 3 ) 15. (s t 3 )(st) y y 17..5j 3 (3h 5 j 7 ) See Eample See Eample 3 Etra Practice See page EP (3m 3 n )(1 5mn 5 ) 19. 3z(5z z) 0. 3h (6h 3h 3 ) 1. 3cd(c 3 d cd ). b(b 7b 10) 3. 3s t (s t 5st s t ). A rectangle has a base of length 3 y and a height of 3 y 3. Write and simplify an epression for the area of the rectangle. Then find the area of the rectangle if and y 1. PRACTICE AND PROBLEM SOLVING Multiply. 5. (3b )(8b ) 6. (m n)(mn ) 7. (a b )(3ab ) 8. 7g(g 5) 9. 3m (m 3 5m) 30. ab(3a b 3ab ) 31. ( 3 y 5 ) 3. m( 3) 33. f g (3 f g 3 ) 3. ( 9) 35. (m p )(5m p 3mp 3 6m p) 36. 3wz(5w z wz 6w z ) 37. Feli is building a cylindrical-shaped storage container. The height of the container is 3 y 3. Write and simplify an epression for the volume using the formula V πr h. Then find the volume with r 1 1 feet, 3, and y Chapter 1 Polynomials

30 38. Health The table gives some formulas for finding the target heart rate for a person of age a eercising at p percent of his or her maimum heart rate. Target Heart Rate Male Female Nonathletic p(0 a) p(6 a) Fit 1 p(10 a) 1 p( a) a. Use the Distributive Property to simplify each epression. b. Use your answer from part a to write an epression for the difference between the target heart rate for a fit male and for a fit female. Both people are age a and are eercising at p percent of their maimum heart rates. 39. What s the Question? A square prism has a base area of and a height of 3. If the answer is 3 3, what is the question? If the answer is 1 16, what is the question? 0. Write About It If a polynomial is multiplied by a monomial, what can you say about the number of terms in the answer? What can you say about the degree of the answer? 1. Challenge On a multiple-choice test, if the probability of guessing each question correctly is p, then the probability of guessing two or more correctly out of four is 6p (1 p p ) p 3 (1 p) p. Simplify the epression. Then write an epression for the probability of guessing fewer than two out of four correctly. AF1.3, MG.1. Multiple Choice The width of a rectangle is 13 feet less than twice its length. Which of the following shows an epression for the area of the rectangle? A 13 B 13 C 13 D Short Response A triangle has base 10cd and height 3c d cd. Write and simplify an epression for the area of the triangle. Then evaluate the epression for c and d 3. Combine like terms. (Lesson 3-). 8 3y 7 5. m n 7 n a 11 6b 10a 7b Find the surface area of each figure. Use 3.1 for π. (Lesson 10-) 7. a rectangular prism with base in. by 3 in. and height.5 in. 8. a cylinder with radius 10 cm and height 7 cm 1-5 Multiplying Polynomials by Monomials 615

31 Multiply Binomials 1-6 Use with Lesson 1-6 KEY REMEMBER The area of a rectangle with base b and height h is given by A bh. California Standards KEYWORD: MT8CA Lab1 Preview of Algebra Students add, subtract, multiply, and divide monomials and polynomials. Students solve multistep problems, including word problems, by using these techniques. 1 1 You can use algebra tiles to find the product of two binomials. Activity 1 1 To model ( 3)( 1) with algebra tiles, make a rectangle with base 3 and height Area ( 3)( 1) 7 3 Use algebra tiles to find ( )( 1). 1 Area ( )( 1) 3 Think and Discuss 1. Eplain how to determine the signs of each term in the product when you are multiplying ( 3)( ).. How can you use algebra tiles to find ( 3)( 3)? 616 Chapter 1 Polynomials

32 Try This Use algebra tiles to find each product. 1. ( )( ). ( 3)( ) 3. ( 5)( 3) Activity Write two binomials whose product is modeled by the algebra tiles below, and then write the product as a polynomial epression. The base of the rectangle is 5 and the height is, so the binomial product is ( 5)( ). The model shows one -tile, seven -tiles, and ten 1-tiles, so the polynomial epression is Think and Discuss 1. Write an epression modeled by the algebra tiles below. How many zero pairs are modeled? Describe them. Try This Write two binomials whose product is modeled by each set of algebra tiles below, and then write the product as a polynomial epression Hands-On Lab 617

33 1-6 Multiplying Binomials California Standards Preview of Algebra Students add, subtract, multiply, and divide monomials and polynomials. Students solve multistep problems, including word problems, by using these techniques. Vocabulary FOIL Why learn this? You can multiply binomials to determine the area of a walkway around a cactus garden. (See Eample.) You can use the Distributive Property to multiply two binomials. 678 ( y)( z) ( z) y( z) First z y yz The product can be simplified using the FOIL method: multiply the First terms, the Outer terms, the Inner terms, and the Last terms of the binomials. Last F O I L Inner Outer EXAMPLE 1 When you multiply two binomials, you will get four products. Then combine like terms. Multiplying Two Binomials Multiply. (p )(3 q) (m n)(p q) (p )(3 q) FOIL (m n)(p q) 3p pq 6 q mp mq np nq ( )( 5) (3m n)(m n) ( )( 5) FOIL (3m n)(m n) m 6mn mn n 7 10 Combine like terms. 3m 5mn n 618 Chapter 1 Polynomials

34 EXAMPLE Landscaping Application Find the area of a bark walkway of width ft around a 1 ft by 5 ft cactus garden. Area of Area of Walkway Total Area Flower Bed (5 )(1 ) (5)(1) The walkway area is 3 ft. Binomial products of the form (a b), (a b), and (a b)(a b) are often called special products. EXAMPLE 3 Special Products of Binomials Multiply. ( 3) (a b) ( 3)( 3) (a b)(a b) a ab ab b 6 9 a ab b (n 3)(n 3) (n 3)(n 3) n 3n 3n 3 n 9 3n 3n 0 Special Products of Binomials (a b) a ab ab b a ab b (a b) a ab ab b a ab b (a b)(a b) a ab ab b a b Think and Discuss 1. Give an eample of a product of two binomials that has terms, one that has 3 terms, and one that has terms. 1-6 Multiplying Binomials 619

35 1-6 Eercises California Standards Practice Preview of Algebra KEYWORD: MT8CA 1-6 KEYWORD: MT8CA Parent GUIDED PRACTICE See Eample 1 Multiply. 1. ( 5)(y ). ( 3)( 7) 3. (3m 5)(m 9). (h )(3h ) 5. (m )(m 7) 6. (b 3c)(b c) See Eample See Eample 3 7. A courtyard is 0 ft by 30 ft. There is a walkway of width all the way around the courtyard. Find the area of the walkway. Multiply. 8. ( ) 9. (b 3)(b 3) 10. ( ) 11. (3 5) See Eample 1 INDEPENDENT PRACTICE Multiply. 1. ( )( 3) 13. (v 1)(v 5) 1. (w 6)(w ) 15. (3 5)( 6) 16. (m 1)(3m ) 17. (3b c)(b 5c) 18. (3t 1)(t 1) 19. (3r s)(r 5s) 0. (5n 3b)(n b) See Eample See Eample 3 1. Construction The Gonzalez family is having a pool to swim laps built in their backyard. The pool will be 5 yards long by 5 yards wide. There will be a cement deck of width yards around the pool. Find the total area of the pool and the deck. Multiply.. ( 5) 3. (b 3). ( )( ) 5. ( 3)( 3) 6. ( 1) 7. (a 7) Etra Practice See page EP5. PRACTICE AND PROBLEM SOLVING Multiply. 8. (m 6)(m 6) 9. (b 5)(b 1) 30. (q 6)(q 5) 31. (t 9)(t ) 3. (g 3)(g 3) 33. (3b 7)(b ) 3. (3t 1)(6t 7) 35. (m n)(m 3n) 36. (3a 6b) 37. (r 5)(r 5) 38. (5q ) 39. (3r s)(5r s) 0. A metalworker makes a bo from a 15 in. by 0 in. piece of tin by cutting a square with side length out of each corner and folding up the sides. Write and simplify an epression for the area of the base of the bo. 60 Chapter 1 Polynomials

36 Life Science A. V. Hill ( ) was a biophysicist and pioneer in the study of how muscles work. He studied muscle contractions in frogs and came up with an equation relating the force generated by a muscle to the speed at which the muscle contracts. Hill epressed this relationship as (P a)(v b) c, where P is the force generated by the muscle, a is the force needed to make the muscle contract, V is the speed at which the muscle contracts, b is the smallest contraction rate of the muscle, and c is a constant. 1. Use the FOIL method to simplify Hill s equation.. Suppose the force a needed to make the muscle contract is approimately 1 the maimum force the muscle can generate. Use Hill s equation to write an equation for a muscle generating the maimum possible force M. Simplify the equation. 3. Write About It In Hill s equation, what happens to V as P increases? What happens to P as V increases? (Hint: You can substitute the value of 1 for a, b, and c to help you see the relationship between P and V.). Challenge Solve Hill s equation for P. Assume that no variables equal 0. The muscles on opposite sides of a bone work as a pair. Muscles in pairs alternately contract and rela to move your skeleton. AF1.3, AF.1 5. Multiple Choice Which polynomial shows the result of using the FOIL method to find ( )( 6)? A 1 B 6 1 C 1 D 6. Gridded Response Multiply 3a b and 5a 8b. What is the coefficient of ab? Solve. (Lesson -8) j y Simplify. (Lesson 1-) 51. (m 3m 6) 5. 3(a b a 3ab) ab 53. y (y 3 y y) 1-6 Multiplying Binomials 61

37 Quiz for Lessons 1-3 Through Adding Polynomials Add. 1. ( ) ( 6). (30 7) (1 5) 3. (7b 3 c 6b c 3bc) (8b 3 c 5bc 13) (b c 5bc 9). (w w 6) (3w w 5) (w ) 5. Each side of an equilateral triangle has length w. Each side of a square has length 3w. Write an epression for the sum of the perimeter of the equilateral triangle and the perimeter of the square. 1- Subtracting Polynomials w 3w Find the opposite of each polynomial y 3 7. m 6m v 7v Subtract b (3b 6b 8) 10. (13a a ) (9 a 7a) 11. (6 6) (3 7) 1. The population of a bacteria colony after h hours is h 3 5h h 00. The population of another bacteria colony is 3h 3 h 5h 00. Write an epression to show the difference between the two populations. Evaluate the epression for h Multiplying Polynomials by Monomials Multiply. 13. ( 3 y 3 )(3y 6 ) 1. (3hj 5 )(6h j 5 ) 15. s t (s t 3st s t ) 16. A triangle has a base of length y and a height of 3 y. Write and simplify an epression for the area of the triangle. Then find the area of the triangle if and y Multiplying Binomials Multiply. 17. ( )( 6) 18. (3m )(m 8) 19. (n 5)(n 3) 0. ( 6) 1. ( 5)( 5). (3 )(3 ) 3. A rug is placed in a 10 ft 0 ft room so that there is an uncovered strip of width all the way around the rug. Find the area of the rug. 6 Chapter 1 Polynomials

38 Cooking Up a New Kitchen Javier is a contractor who remodels kitchens. He drew the figure to help calculate the dimensions of a countertop surrounding a sink that is inches long and y inches wide. in. 1. Write a polynomial that Javier can use to find the perimeter of the outer edge of the countertop. 6 in. in. y 6 in.. Someone orders a countertop for a sink that is 18 inches long and 1 inches wide. Javier puts tape around the outer edge of the countertop to protect it while it is being moved. Use the polynomial to determine how many inches of tape are needed. 3. Write a polynomial that Javier can use to find the area of the countertop for any size sink.. The marble for the countertop costs $1.5 per square inch. Write a polynomial that gives the cost of the countertop. 5. Find the cost of the countertop for the 18-inch by 1-inch sink. Eplain your answer. Concept Connection 63

39 Short Cuts You can use properties of algebra to eplain many arithmetic shortcuts. For eample, to square a two-digit number that ends in 5, multiply the first digit by one more than the first digit, and then place a 5 at the end. To find 35, multiply the first digit, 3, by one more than the first digit,. You get 3 1. Place a 5 at the end, and you get 15. So Why does this shortcut work? You can use FOIL to multiply 35 by itself: (30 5)(30 5) First use the shortcut to find each square. Then use FOIL to multiply the number by itself Can you eplain why the shortcut works? Use FOIL to multiply each pair of numbers Write a shortcut for multiplying two-digit numbers with a first digit of 1. Rolling for Tiles For this game, you will need a number cube, a set of algebra tiles, and a game board. Roll the number cube, and draw an algebra tile: 1,, 3,, 5, 6. The goal is to model epressions that can be added, subtracted, multiplied, or divided to equal the polynomials on the game board. A complete set of rules and a game board are available online. KEYWORD: MT8CA Games 6 Chapter 1 Polynomials

40 Materials 3 sheets of decorative paper ruler compass scissors glue markers PROJECT Polynomial Petals A Pick a petal and find a fact about polynomials! Directions 1 Draw a 5-inch square on a sheet of decorative paper. Use a compass to make a semicircle on each side of the square. Cut out the shape. Figure A 3 5 Draw a 3 1 -inch square on another sheet of decorative paper. Use a compass to make a semicircle on each side of the square. Cut out the shape. Draw a 1 -inch square on the last sheet of decorative paper. Use a compass to make a semicircle on each side of the square. Cut out the shape. Glue the medium square onto the center of the large square so that the squares are at a 5 angle to each other. Figure B Glue the small square onto the center of the medium square in the same way. B Taking Note of the Math Write eamples of different types of polynomials on the petals. Then use the remaining petals to take notes on the key concepts from the chapter. When you re done, fold up the petals. 65

41 Vocabulary binomial degree of a polynomial FOIL polynomial trinomial Complete the sentences below with vocabulary words from the list above. 1. The epression is an eample of a? whose? is 3.. Use the? method to find the product of two?. 3. A polynomial with terms is called a?. A polynomial with 3 terms is called a?. 1-1 Polynomials (pp ) Preview of 1A10.0; 7AF1. EXAMPLE Classify each epression as a monomial, a binomial, a trinomial, or not a polynomial trinomial y 7 3 y not a polynomial Find the degree of each polynomial. 3 1 degree 3 n 3n 16n degree EXERCISES Classify each epression as a monomial, a binomial, a trinomial, or not a polynomial.. 5t 7t 8 5. r 3r g 7g 3 5 g 7. a b 3 c 5 8. y 9. 6st 7s Find the degree of each polynomial r 9r m m m Chapter 1 Polynomials

42 1- Simplifying Polynomials (pp ) Preview of 1A10.0; 7AF1.3 EXAMPLE Simplify ( 7) 5 EXERCISES Simplify. 15. t 6t 3t t 7t gh 5g h 7gh g h 17. (5mn 3m) 18. (a b) 6b 19. 5(st 6t) 16st 7t Adding Polynomials (pp ) Preview of 1A10.0; 7AF1.3 EXAMPLE Add. (3 ) (5 3 ) Identify like terms. 8 Combine like terms. (8t 3 t 6) (t 7t ) 8t 3 t 6 Place like terms t 7t in columns. 8t 3 t 3t Combine like terms. EXERCISES Add. 0. ( 3 7) ( 5 1) 1. (5 3 ) ( 5 9). (5h 5) (h 3) (3h 1) 3. (3y 5 y y) (3 y 6y y ). (n 6) (3n ) (8 6n ) 1- Subtracting Polynomials (pp ) Preview of 1A10.0; 7AF1.3 EXAMPLE Subtract. (6 5) (7 8 ) 6 5 (7 8 ) Add the opposite Associative Property 3 Combine like terms. EXERCISES Subtract. 5. ( ) ( 5 ) 6. (w w 6) (w 8w 8) 7. (3 8 9) (7 8 5) 8. (ab 5ab 7a b) (3a b 6ab) 9. (3p 3 q p q ) (pq p 3 q ) Study Guide: Review 67

43 1-5 EXAMPLE Multiply. (3 y 3 )(y ) (3 y 3 )(y ) Multiplying Polynomials by Monomials (pp ) 3 1 y y 5 Multiply the coefficients and add the eponents. EXERCISES Preview of 1A10.0; 7AF1., 7AF1.3, 7AF. Multiply. 30. (st 3 )(s 3st 8) 31. 6a b(a b 5ab 6a b) 3. m(m 8m 1) 33. 5h(3gh g 3 h 6h g) (ab )(a b 3ab 6a 8) 3. 1 j 3 k (j k 3jk j 3 k 3 ) y 5 (5 y y 9 8y y ) (ab )(a b 3ab 6a 8) 8a 3 b 6a b 3 1a b 16ab 1-6 Multiplying Binomials (pp ) Preview of 1A10.0 EXAMPLE Multiply. (r 8)(r 6) (r 8)(r 6) FOIL r 6r 8r 8 Combine like terms. r r 8 (b 6) (b 6)(b 6) FOIL EXERCISES Multiply. 36. (p 6)(p ) 37. (b )(b 6) 38. (3r 1)(r ) 39. (3a b)(a 5b) 0. (m 7) 1. (3t 6)(3t 6). (3b 7t)(b t) 3. (10 3)( ). ( y 11) b 6b 6b 36 b 1b 36 Combine like terms. 68 Chapter 1 Polynomials

44 Classify each epression as a monomial, a binomial, a trinomial, or not a polynomial. 1. t t a 3 b 6 3. m 5m 8 Find the degree of each polynomial.. 6 9b m y 7. The volume of a cube with side length is given by the polynomial What is the volume of the cube if 3? Simplify. 8. a b 5b 6a b 9. 3( 6 10) 10. y 3y y y 11. 6(b 7b) 3b 5b 1. The area of one face of a cube is given by the epression s 9s. Write a polynomial to represent the total surface area of the cube. Add. 13. ( 1) ( 5) 1. (1 5) (9 5) 15. (3bc b c 5bc ) (bc bc ) 16. (6h 5 3h 3 h 6 ) (h 6 h 5h ) 17. (b 3 c 8b c 5bc) (6b 3 c bc 3) (b c 3bc 11) 18. Harold is placing a mat of width w around a 16 in. by 0 in. portrait. Write an epression for the perimeter of the outer edge of the mat. Subtract. 19. (m n 5mn mn ) (mn m n) 0. (1a a ) (6 a 8a) 1. (3a b 5a b 6ab ) (a b 7a b). ( j 7j j) (5j 3 j 6j 1) 3. A circle whose area is 3 is cut from a rectangular piece of plywood with area 3 1 and discarded. Write an epression for the area of the remaining plywood. Evaluate the epression for in. Multiply.. (3)(5 ) 5. ( y)(5y 3 ) 6. (a b )(5a b 5 ) 7. a(a 3 a 5) 8. 3m 3 n (m 3 n 5m n ) 9. 3a 3 (ab ab 8a) 30. ( )( 1) 31. ( )( ) 3. (a 3)(a 7) 33. A student forms a bo from a 10 in. by 15 in. piece of cardboard by cutting a square with side length out of each corner and folding up the sides. Write and simplify an epression for the area of the base of the bo. Chapter 1 Test 69