Chapter 7 Multiplying and Dividing Polynomials PDF

CHAPTER 7 Multiplying and Dividing Polynomials Polynomials have been used for centuries to explore and describe relationships. These expressions are useful in everyday life. For example, a landscape designer might use a polynomial to help with calculating the cost of a landscapin...

CHAPTER 7 Multiplying and Dividing Polynomials Polynomials have been used for centuries to explore and describe relationships. These expressions are useful in everyday life. For example, a landscape designer might use a polynomial to help with calculating the cost of a landscaping project. The landscaper can calculate a number of different possible costs using a polynomial expression. This saves a lot of time. For what other jobs might polynomials be helpful? What You Will Learn to multiply and divide polynomials using models and symbols to simplify polynomial expressions by combining like terms 250 Chapter 7 NEL  Key Words monomial binomial polynomial distributive property Literacy Link A spider map can help you understand and connect new terms and concepts. This spider map is designed to be used throughout the chapter. Create a spider map in your math journal or notebook. As you work through the chapter, complete the map. After completing section 7.1, beside each heading on the upper left and upper right legs, provide your own examples and methods for multiplying monomials and dividing monomials. After completing section 7.2, beside each heading on the lower left leg, provide your own examples and methods for multiplying polynomials by monomials. After completing section 7.3, beside each heading on the lower right leg, provide your own examples and methods for dividing polynomials by monomials. Mult omials Mon ls Mon iding omia iplyi Div ng Multiplying And Dividing Polynomials ials ials Divid y Monom m by M g Polyno b ing P onom olyn als n iplyi omia i Mult ls NEL Chapter 7 251  FOLDABLES TM Study Tool Making the Foldable Step 3 Materials Stack three sheets of grid paper so that the bottom sheet of 11 × 17 paper edges are 2.5 cm apart. Fold the top edge of the sheets two sheets of 8.5 × 11 paper and align the edges so that all tabs are the same size. scissors Staple along the fold. Label as shown. three sheets of 8.5 × 11 grid paper ruler Multiplying and Dividing stapler Polynomials by Monomials 7.2 Multiply a Polynomial by a Step 1 Monomial Using an Area Model 7.2 Multiply a Polynomial by a Monomial Using Algebra Tiles Fold the long side of a sheet of 11 × 17 paper in half. 7.2 Multiply a Polynomial by a Pinch it at the midpoint. Fold the outer edges of the Monomial Using Symbols 7.3 Divide a Polynomial by a paper to meet at the midpoint. Label it as shown. Monomial Using a Model 7.3 Divide a Polynomial by a Monomial Using Symbols Step 4 Staple the three booklets you made into the Foldable from Step 1 as shown. 7.1 Multiplying and Dividing 7.1 Multiply Polynomials by Monomials Divide Monomials 7.2 Multiply a Polynomial by a Monomials Using a Monomial Using an Area Model Using a Model 7.2 Multiply a Polynomial by a Monomial Using Algebra Tiles Model Step 2 7.2 Multiply a Polynomial by a Monomial Using Symbols Fold the short side of a sheet of 8.5 × 11 paper in half. 7.1 7.3 Divide a Polynomial by a 7.1 Monomial Using a Model Fold in half the opposite way. Make a cut as shown Multiply Divide 7.3 Divide a Polynomial by a through one thickness of paper, forming a two-tab Monomials Monomial Using Symbols Monomials Using Key Words: Using book. Repeat Step 2 to make another two-tab book. monomial Label them as shown. Symbols polynomial Symbols binomial distributive property 7.1 7.1 Multiply Divide Monomials Monomials Using the Foldable Using a Using a Model As you work through the chapter, write the Key Words Model in the remaining space in the centre panel, and provide definitions and examples. Beneath the tabs in the left, 7.1 7.1 right, and centre panels, provide examples, show work, Multiply Divide and record main concepts. Monomials Monomials Using Using On the front of the right flap of the Foldable, record Symbols Symbols ideas for the Wrap It Up! On the back of the Foldable, make notes under the heading What I Need to Work On. Check off each item as you deal with it. 252 Chapter 7 NEL Math Link Landscape Design Gardeners and landscapers are often required to calculate areas when designing a landscape for a backyard, commercial property, or park. When determining how much soil, sand, gravel, mulch, and seed they need for a project, landscape designers also calculate volumes. Here is a landscape design created for a property. Herb Garden 8.5 m Driveway 3m 2m Patio 2.3 m 27 m House Pool 4 m 4m 9m 36 m 1. a) The circular herb garden has a radius of 4.5 m. What is the area of the herb garden? b) If the herb garden must have soil that is 0.5 m deep, what volume of soil is needed? c) What is the difference between the units used to measure the area and the units used to measure the volume of the herb garden? 2. The house is square. What fraction of the property does the house take up? Show two ways to express the answer. 3. The pool is in the shape of a square with a trapezoid attached to it. For the water in the pool to have a depth of 1.7 m, what volume of water is needed? Describe how you calculated the volume. 4. a) The patio has a surface area of 18 m2. If it takes 48 paving stones to cover 1 m2, how many paving stones are needed for the patio? b) Does your answer need to be exact? Explain. 5. What total area is grass? Explain how you calculated the area. In this chapter, you will explore how to multiply and divide polynomials to help you create a landscape design for a park. What types of materials will you use? NEL Math Link 253  7.1 Multiplying and Dividing Monomials Focus on… After this lesson, you will be able to… multiply a monomial by a monomial divide a monomial by a monomial The band council would like to design a Medicine Wheel Did You Know? similar to the one shown above for a square area of the new The Medicine Wheel school courtyard. According to the design, the edges of 2x represents harmony the outer circular pathway will just touch the edges of the and connection. It is square. The outer radius of the circle can be represented an important symbol of the peaceful by 2x. How could you determine the relationship between relationships among the area of the circle and the area of the square? all living things. Explore Multiplying and Dividing Monomials 1. What is the side length of the square in terms of x? Literacy Link A monomial has one 2. a) Write an expression for the area of the circle. term. For example, 5, n4 b) Write an expression for the area of the square. 2x, 3s2, -8cd, and __ 3 are all monomials. Do not use an 3. Show how to compare the two areas using approximate value for a ratio expressed in lowest terms. π. Leave π in the ratio. 4. How does the area of the square compare to the area of the circle? Reflect and Check 5. Would this relationship be the same for any circle inscribed in a square? Explain. 6. a) How would you multiply the monomials 4x and 3x? b) How would you divide the monomial 10x2 by the monomial 5x? 254 Chapter 7 NEL Link the Ideas Example 1: Multiply Monomials Determine each product. a) (5x)(2x) b) (3x)(2y) Solution a) Method 1: Use a Model You can use algebra You can use x-tiles and x2-tiles to model (5x)(2x). tiles to model algebraic expressions. positive x-tile positive y-tile positive xy-tile Each square has an area of (x)(x) = x2. There are 10 positive x2-tiles. So, (5x)(2x) = 10x2. Method 2: Algebraically Multiply the numerical coefficients. positive x2-tile Then, multiply the variables. The same tiles in (5x)(2x) white represent How can you use the = (5)(2)(x)(x) exponent laws to help you negative quantities. = 10x2 multiply the variables? b) Method 1: Use a Model You can use x-tiles, y-tiles, and xy-tiles to model (3x)(2y). Each grey rectangle has an area of (x)(y) = xy. There are 6 positive xy-tiles. So, (3x)(2y) = 6xy. Method 2: Algebraically (3x)(2y) = (3)(2)(x)(y) = 6xy Show You Know Determine each product in two different ways. a) (4x)(2y) b) (-x)(7x) NEL 7.1 Multiplying and Dividing Monomials 255  Example 2: Apply Monomial Multiplication 4.3x What is an expression for the area of the rectangle? 2x Solution You can calculate the area, A, of a rectangle by multiplying the length by the width. A = (4.3x)(2x) A = (4.3)(2)(x)(x) A = (8.6)(x2) A = 8.6x2 An expression for the area of the rectangle is 8.6x2. Show You Know Calculate each product. a) (11a)(2b) b) (-5x)(3.2) Example 3: Divide Monomials Determine each quotient. -10x2 8xy a) ______ b) ____ 2x 4x Solution Strategies a) Method 1: Use a Model Model It You can divide using algebra tiles. -10x2 by representing the numerator with 10 negative x2-tiles. Model ______ 2x Arrange the 10 tiles into a rectangle so that one of the sides is 2 x-tiles long. ? The unknown side length of the rectangle is made up of 5 negative x-tiles. -10x = -5x ______ 2 2x 256 Chapter 7 NEL  Method 2: Algebraically You can divide the numerator and the denominator by 2x. -10x ______ 2 2x How can you use the -5x exponent rules to help -10x 2 you divide the variables? = ______ 2x 1 = -5x b) Method 1: Use a Model 8xy Model ____ by representing the numerator with 8 xy-tiles. Arrange 4x the 8 tiles into a rectangle so that one of the sides is 4 x-tiles long. ? The unknown side length of the rectangle is made up of 2 y-tiles. 8xy ____ = 2y 4x Method 2: Algebraically Divide common factors in the numerator and denominator. 8xy ____ 4x 2 8xy = ____ 4x 1 = 2y Show You Know Determine each quotient. 12xy -14x 2 a) _____ b) ______ 3y -2x NEL 7.1 Multiplying and Dividing Monomials 257  Example 4: Apply Monomial Division The area of a triangle is given by the expression 18x2. The base of the triangle is represented by 4x. What is the height of the triangle in terms of x? Strategies Solution Draw a Diagram Area = 18x2 4x The area of a triangle can be calculated by multiplying the base by the height, then dividing by 2. Area = base × height ÷ 2 So, if the area and base are known, then Height = 2 × ____area base (2)(18x2) Height = ________ 4x Height = 36x _____ 2 4x Divide the numerical coefficients. Then, divide the variables. Height = 9x The height of the triangle is 9x. Show You Know Calculate each quotient. 18x2 a) _____ 3x b) 14y ÷ (-2) -18.6mn c) _________ -3n 258 Chapter 7 NEL  Key Ideas You can represent the multiplication and division of monomials using a model. (2x)(-3x) There are 6 negative x2-tiles. (2x)(-3x) = -6x2 8xy ___ The unknown side length of the 2x rectangle is made up of 4 positive y-tiles. 8xy ___ = 4y ? 2x To multiply monomials algebraically, you can multiply the numerical coefficients and use the exponent rules to multiply the variables. To divide monomials algebraically, you can divide the numerical coefficients and use the exponent rules to divide the variables. Check Your Understanding Communicate the Ideas 1. Explain to a partner at least two ways you could find the product of (3x) and (5x). 2. Laurie used the following method to divide 16n2 by 2n. 16n2 ____ 2n = (16 - 2)(n2 - n) = 14n Does Laurie’s method have any errors? If so, what are her errors and what is the correct solution? NEL 7.1 Multiplying and Dividing Monomials 259 Practise For help with #3 to #8, refer to Example 1 on page 255. b) c) 3. What multiplication statement is represented by each set of algebra tiles? a) 5. Model and complete each multiplication statement. a) (2x)(4x) b) (-4x)(2x) b) c) (-4x)(-2x) d) (-2x)(4x) 6. Represent each multiplication statement with a model. Then, give the product. a) (3x)(5x) b) (x)(-6x) c) (-3x)(2x) d) (-x)(x) For help with #7 to #10, refer to Example 2 on page 256. c) 7. Find the product of each pair of monomials. a) (2y)(5y) b) (3a)(-6b) c) (-q)(-9q) d) ( __32x )(3x) e) (-3r)(-2t) f) (1.5p)(-3p) 8. Multiply each pair of monomials. 4. Determine the multiplication statement a) (3n)(2n) b) (-4k)(-7k) shown by each set of algebra tiles. a) c) (-4w)(2.5w) d) ( ___ 5 ) -3 x (15x) e) (8m)(-0.5n) f) (t)(-7t) 9. A rectangle has a width of 3.9x and a length of 5x. What is an expression for the area of the rectangle? 10. A parallelogram has a base of 8.1z and a height of 4.2z. What is an expression for the area of the parallelogram? 4.2z 8.1z 260 Chapter 7 NEL For help with #11 to #16, refer to Example 3 on b) ? pages 256-257. 11. Write the division statement represented by each set of algebra tiles. a) ? c) b) ? ? c) 13. Model and complete each division. 8x 2 5xy a) ____ b) ____ 2x 5y -12x 2 c) ______ 2x2 d) ____ ? 4x -x 14. Model and complete each division. -15x 2 10xy a) ______ b) _____ 3x 2x 12x 2 c) _____ -9x 2 d) _____ 12. Determine the division statement shown -3x -3x by each set of algebra tiles. a) 15. Find the quotient of each pair of monomials. 7x a) ____ 2 25st b) ____ 125t c) _____ x 5s 5 -8m 81rs 4.5p2 d) _____ e) ____ f) _____ -2m 3rs -3p ? 16. Divide. a) 12.4x2 ÷ x b) -15r ÷ (-4r) c) 0.6t ÷ 0.2t 2 2 d) -18pn ÷ 3n e) k ÷ 4k 2 f) __x2 ÷ 2x 3 NEL 7.1 Multiplying and Dividing Monomials 261 Apply 21. The diagram shows that x is the radius of the large circle and the diameter of the small 17. Find an expression for the area of each circle. Write the ratio of the area of the large figure. circle to the area of the small circle. Simplify a) the expression. 3x 11x b) x 4p 10p c) _1 w 22. A circle is inscribed in a square as shown. 2 r 18. What is the missing dimension in each figure? a) b) A = 3.3w2 In terms of the radius, r, determine each 1.1w of the following ratios. A = 15x2 a) the area of the square to the area of 5x the circle b) the perimeter of the square to the circumference of the circle 19. The area of a rectangle is 72d2 and its length is 20d. What is an expression for its width? Literacy Link An inscribed circle fits exactly into another figure so that the edges of the two figures touch, but do not intersect. 20. Claire wants to build a patio outside her café. The rectangular space outside of Claire’s café is three times as long as it is 23. Jonasie and Elisa wide. The area of the space is 48 m2. Claire are taking two would like to build a patio with dimensions tourists on a trip to 3.5 m by 12.5 m in this space. Will it fit? photograph caribou. Explain. The visitors will be travelling by dogsled. The dogsled’s length is 4 times its width. The sled has a rectangular base area of 3.2 m2. The equipment to be loaded on the sled measures 0.8 m wide by 3.5 m long. Will the equipment fit on the sled as it is presently packed? Explain your answer. 262 Chapter 7 NEL Extend 27. A contractor needs to order the glass for a window. The window is in the shape of an 24. A rectangular prism has a volume in cubic isosceles triangle and the height of the centimetres expressed as the monomial window is 2.5 times the base width. 60xy. The length and width of the prism, in centimetres, are 4x and 3y respectively. a) Determine the height. b) Write an expression for the surface area of the rectangular prism. 9n 25. How is determining ____2 similar to and 3n 9n2 ? a) Determine an expression for the area different from determining ____ 3n of the window in terms of the width of its base. 26. A and B represent monomials. When A is b) If the width of the base must be 85 cm, multiplied by B, the result is A. When A is what is the largest window area the divided by B, the result is A. What is the contractor can use? monomial B? Landscape designs for gardens may include rectangular and circular areas for flower beds, lawns, patios, and pools. As a landscape designer, you sometimes need to: calculate the volume of material, such as soil, gravel, water, or mulch, needed to fill these areas to a certain depth calculate the area that a known volume of material will cover The following are the formulas for these calculations: Volume = area × depth volume Area = _______ depth Draw a rectangle and circle that might be used in landscaping. Label the You may wish to create design element that each shape represents. a spreadsheet that Use variables for the dimensions of the shapes. allows you to enter the Create an area formula for each shape. values to calculate For each shape, tell what type of material you will use to fill it. areas and volumes. Also, tell what the depth of the material will be. Create a volume formula for each shape. Along with each formula, include an explanation concerning any coefficients How do these you use. For example, you may have to convert centimetre measurements coefficients relate to your landscape design? to metres. NEL 7.1 Multiplying and Dividing Monomials 263  7.2 Multiplying Polynomials by Monomials Focus on… After this lesson, you will be able to… multiply a polynomial by a monomial In the 1880s, braking a train was a tough job. The engineer “whistled down” for brakes and reversed his engine. Two people were responsible for the brakes. One rode in the cab on the engine and the other rode in the caboose. These people would run toward each other over the swaying, rocking car tops, tightening each car’s brake control wheel as they went. Physics Link Air brakes on trains became commonplace in the 1900s, making the task of stopping a train much easier. Air brakes use compressed air in a piston to push the brake shoe onto the train wheel. The system works like a bicycle brake. 264 Chapter 7 NEL Explore Multiplying a Polynomial by a Monomial Literacy Link When a train’s brakes are applied, the train travels a distance before it A polynomial is made stops. After t seconds, the distance, in metres, that the train travels is up of terms connected given by the polynomial 2t(20 - t). by addition or subtraction. 1. What part of the diagram does 2t(20 - t) represent? Examples: 20 x+5 t 2d - 2.4 x 3s2 + 5s - 6 h2 __ 2t __ -h 2 4 2. What polynomial represents the unknown length in the diagram? How did you determine this polynomial? 3. Find three rectangles in the diagram. What is an expression for the area of the largest rectangle? What is an expression for the area of the smallest rectangle? 4. What is the difference in area between the largest and smallest rectangles? Show two ways to find your answer. 5. Calculate the area of the medium-sized rectangle using the dimension you determined in #2. Reflect and Check 6. Describe the steps you used in #5 to calculate the area of the medium-sized rectangle. 7. How is the area of the medium-sized rectangle related to the areas of the large rectangle and the small rectangle? 8. How far does the train travel in 10 s? Show how you arrived at your answer. NEL 7.2 Multiplying Polynomials by Monomials 265  Link the Ideas Example 1: Multiply a Polynomial by a Monomial Using an Area Model Determine the product. (3x)(2x + 4) Solution Strategies Draw a rectangle with side dimensions that represent 3x and 2x + 4. Draw a Diagram 2x + 4 2x 4 3x A1 A2 Calculate the area of each rectangle. A1 = (3x)(2x) A1 = 6x2 A1= 6x2 A2= 12x A2 = (3x)(4) A2 = 12x The total area is A1 + A2 = 6x2 + 12x. Show You Know Calculate each product. a) (2x)(x + 3) b) (2 + c)(c) Example 2: Multiply a Polynomial by a Monomial Using Algebra Tiles Find the product. (2x)(3x - 5) Solution You can use x-tiles and negative 1-tiles to model 2x and 3x - 5. 266 Chapter 7 NEL  Use x2-tiles and negative x-tiles to model (2x)(3x - 5). There are 6 positive x2-tiles and 10 negative x-tiles. So, (2x)(3x - 5) = 6x2 - 10x. Show You Know Find each product. a) (2 + 3x)(3x) b) (4x)(2x - 1) Example 3: Multiply a Polynomial by a Monomial Algebraically The dimensions of a rectangular gym floor are represented by the expressions 4x and 5x - 3. What is a polynomial expression for the area 4x of the gym floor? Write the expression in simplified form. 5x - 3 Solution You can calculate the area of a rectangle, A, by multiplying the width by the length. Literacy Link The distributive A = (4x)(5x - 3) property allows you to expand algebraic You can apply the distributive property. Multiply 4x by each term expressions. Multiply in the binomial 5x - 3. the monomial by each term in the polynomial. A = (4x)(5x - 3) a(b + c) = ab + ac A = (4x)(5x) - (4x)(3) A = (4)(5)(x)(x) -(4)(3)(x) A = 20x2 - 12x Literacy Link A binomial is a A simplified expression for the area of the gym floor is 20x2 - 12x. polynomial with two terms, such as 6y2 + 3 and 2x - 5. Show You Know Calculate each product. a) (-3x)(2x + 5) b) (5y)(11 - x) NEL 7.2 Multiplying Polynomials by Monomials 267  Key Ideas You can represent the multiplication of a polynomial by a monomial using models. area model algebra tiles 2x + 2 2x 2 3x A1= 6x2 A2= 6x (3x)(2x + 2) The product is represented by A1 + A2. (3x)(2x + 2) = 6x2 + 6x (2x)(-2x + 3) There are 4 negative x2-tiles and 6 positive x-tiles. (2x)(-2x + 3) = -4x2 + 6x To multiply a polynomial by a monomial algebraically, you can expand the expression using the distributive property. Multiply each term of the polynomial by the monomial. (-1.2x)(3x - 7) = (-1.2x)(3x) - (-1.2x)(7) = -3.6x2 + 8.4x Check Your Understanding Communicate the Ideas 1. Describe two methods you could use to multiply polynomials by monomials. 2. Sara is going to simplify the expression (3x)(2x + 4). Can she add the terms in the brackets and then multiply? Explain. 3. Mahmoud used the following method to expand the expression (5x)(2x + 1). (5x)(2x + 1) = 10x2 + 1 a) Show that Mahmoud’s solution is incorrect. b) How would you correct his solution? 268 Chapter 7 NEL Practise For help with #4 to #7, refer to Example 1 on page 266. b) 4. What multiplication statement is represented by each area model? a) 2x 4 b) 4k 3k c) 3x 3.6 c) 3.2k 5.1 k 5. Determine the multiplication statement shown by each area model. a) 3y 7 b) 3.5f 4y f 9. Determine the multiplication statement shown by the algebra tiles. 2 a) c) 7 0.9k 2k 6. Expand each expression using an area model. a) (3.2r + 1)(4r) b) ( __21 a )(3a + 6) 7. Use an area model to expand each expression. a) (2x)(4x + 2) b) (6k + 2)(4.5k) b) For help with #8 to #11, refer to Example 2 on pages 266-267. 8. What multiplication statement is represented by the algebra tiles? a) c) NEL 7.2 Multiplying Polynomials by Monomials 269 10. Expand each expression, using algebra tiles. 15. Lee has decided to build a shed on a square a) (x - 5)(3x) b) (2x)(-2x + 3) concrete slab. The shed has the same width, w, as the slab. Its length is 2 m shorter than the 11. Use algebra tiles to expand each expression. width of the slab. a) (4x + 2)(-3x) b) (-4x)(3x - 1) a) What is an expression for the area of the shed? For help with #12 and #13, refer to Example 3 on b) If the width, w, of the slab is 4 m, what page 267. is the area of the shed? 12. Expand using the distributive property. a) (2x)(3x - 1) b) (3p)(2p - 0.8) c) (0.5m)(7 - 12m) d) ( __21r - 2 )(-r) 2m e) (2n - 7)(8.2) w f) (3x)(x + 2y + 4) 16. The basketball court for the Jeux de la 13. Multiply. Francophonie is 5.5 m longer than 1.5 times a) (4j)(2j - 3) the width. b) (-1.2w)(3w - 7) a) What is an expression for the area of the c) (6x)(4 - 2.4x) basketball court? b) If the length is 28 m, what is the area of d) ( __73 v + 7 )(-1) the basketball court? e) (3 - 9y)(y) f) (-8a - 7b - 2)(8a) Did You Know? The Jeux de la Francophonie are games for Apply French-speaking people. They are held every four years in different locations around the world. The 14. A rectangular Kwakiutl button blanket games include sports and artistic events. Canada is has a width of 3x and a length of 4x - 3. represented by three teams: Québec, New Brunswick, and a third team representing the rest of Canada. Digital rights not available. Digital rights not available. a) What is an expanded expression for the 17. A rectangular field is (4x + 2) m long. The area of the blanket? width of the field is 2 m shorter than the b) What is a simplified expression for the length. What is an expression for the area perimeter of the blanket? of the field? 270 Chapter 7 NEL 18. A rectangular skateboard park is (3x) m 20. The surface area, SA, of a cylinder is long. Its width is 4 m less than the length. SA = 2πr2 + 2πrh, where r is the radius and a) What is an expression for the area of h is the height. The formula for the volume, the park? V, of a cylinder is V = πr2h. What is the surface area of a cylinder that has a height b) If x = 15, what is the area of the park? of 5 cm and a volume of 80π cm3? Extend r 19. A rectangluar packing crate has the h dimensions shown, in metres. 21. A rectangle measuring (12n) m by 8 m has a square of side length 2 m cut out in the 2x + 2 four corners. The cut-out shape forms an open box when the four corners are folded and taped. x a) Draw the box and label its dimensions. 2x b) What is the surface area of this a) What is an expression for the total open box? surface area of the crate? c) What is the capacity of this box? b) What is an expression for the volume of the crate? You are drawing up plans for a landscape design. You are going to include one of the following design elements, which will be in the shape of a rectangle: swimming pool concrete patio hockey rink beach volleyball pit The rectangular shape is 2 m longer than twice the width. You must choose For example, a swimming pool an appropriate depth for your design element. may have a depth of 1.5 m. a) Create a formula for calculating the volume of material needed to fill your design element. b) Use your formula to calculate the volume of material needed for widths Is a pool filled right to the top of 2 m, 3 m, 4 m, and 5 m. Which width would you prefer for your design with water? How might this element? Why? affect your formula? NEL 7.2 Multiplying Polynomials by Monomials 271  7.3 Dividing Polynomials by Monomials Focus on… After this lesson, you will be able to… divide a polynomial by a monomial When you are buying a fish tank, the size of the tank depends on the size and habits of the fish. A tank for a jaguar cichlid, or Parachromis managuensis, should have the minimum dimensions shown, in metres. The volume of the rectangular tank can be represented by the polynomial expression 7.5w2 - 3w. How could you determine a polynomial expression that represents the 0.6 m length of the tank in terms of w? w Materials Explore Dividing a Polynomial by a Monomial algebra tiles A rectangular solid has a width of 2x, a height of 3, and an unknown length. The area of the base of the solid is represented by the polynomial 2x2 + 4x. 1. Show that the volume of the solid shown can be 3 represented by the polynomial 6x2 + 12x. 2x 2. Use algebra tiles to represent the area of the rectangular base. 3. Count the number of x-tiles and 1-tiles required for the missing dimension of the rectangle. What expression represents the missing dimension? Reflect and Check What happens if you multiply your 4. Show that your expression for the missing dimension in #3 is correct. expression by the width of the 5. Describe the steps you would take to find an expression for the rectangular solid? ratio of the volume to the side measuring 2x. 272 Chapter 7 NEL Link the Ideas Example 1: Divide a Polynomial by a Monomial Using a Model Determine the quotient. 6x2 - 8x ________ 2x Solution You can use algebra tiles. ? Strategies Use 6 positive x2-tiles and Model It 8 negative x-tiles to represent the polynomial 6x2 - 8x. The vertical side of the rectangle represents the monomial divisor, 2x. Count the number of positive x-tiles and negative 1-tiles required to complete the horizontal side of the rectangle. There are 3 positive x-tiles and 4 negative 1-tiles, or 3x - 4. This expression represents the result of dividing the polynomial, 6x2 - 8x, by the monomial, 2x. You can also determine 6x - 8x = 3x - 4 2 ________ the quotient 2x algebraically. 6x2 - 8x ________ Check: 2x Multiply the quotient, 3x - 4, by the divisor, 2x. 6x 2 8x = ___ - ___ 2x 2x 3x 4 (2x)(3x - 4) How do you know 6x2 ⁄ ⁄ 8x = ___ - ___ = (2x)(3x) - (2x)(4) that the answer is 2x ⁄ ⁄ 2x 1 1 = 6x2 - 8x correct? = 3x - 4 Show You Know Determine each quotient. 3x2 + 6x 8x - 2x 2 a) ________ b) ________ 3x 2x NEL 7.3 Dividing Polynomials by Monomials 273  Example 2: Dividing a Polynomial by a Monomial Algebraically a) What is the ratio of the surface area to the radius The formula for the r surface area of a of the cylinder? Write the ratio in simplified form. h cylinder is 2πr2 + 2πrh. b) If the height, h, of the cylinder is the same as the radius, r, what is the ratio of the surface area to the radius? Write the ratio in simplified form. Solution surface area 2πr2 + 2πrh a) ___________ = ____________ b) Substituteh = r into the ratio radius r from part a). The expression can be broken surface area ___________ down into two parts. radius surface area ___________ = 2πr + 2πh radius = 2πr + 2π(r) = _____ 2πrh 2πr2 + _____ = 2πr + 2πr r r = 4πr rr 1 = _____ 2πrh 2πr + _____ 2 r r 1 1 = 2πr + 2πh Show You Know Determine each quotient. 15x2 - 12x -2t + 4t 2 a) ___________ b) _________ 3x 2t Key Ideas You can divide a polynomial by a monomial using a model. ? 4x - 6x 2 ________ 2x The unknown side length of the rectangle is made up of 2x - 3 tiles. 4x2 - 6x ________ = 2x - 3 2x When you divide a polynomial by a monomial algebraically, you can divide the numerical coefficients and apply the exponent laws to the variables. 4x - 8x ________ 2 2x 4x2 8x = ___ - ___ 2x 2x 2x 4 8x 4x - ___ 2 = ___ 2x 2x 1 1 = 2x - 4 274 Chapter 7 NEL Check Your Understanding Communicate the Ideas 3x + 6x 2 1. Explain how you would perform the following division: ________. 2x 2. Anita used the following method to simplify an expression: 9k - 3k2 ________ a) Show that Anita’s solution is incorrect. 3 9k ___ 3k 2 b) How would you correct her solution? = - ___ 3 3 = 3k - 1 3. Use a model to show a polynomial division statement with a quotient of 3x + 2. Practise For help with #4 to #7, refer to Example 1 on page 273. 5. Determine the division statement 4. What division statement is represented by represented by the algebra tiles and give the algebra tiles? Determine the quotient. the quotient. a) ? a) ? b) ? b) ? c) ? c) ? NEL 7.3 Dividing Polynomials by Monomials 275  6. Divide each expression, using a model. 11. A rectangular fish tank has the dimensions 5x - 10x 2 shown, in metres. The volume of the tank a) __________ 5x can be represented by 7.5w2 - 3w. 4x + 12x 2 b) __________ 2x 7. Use a model to divide each expression. -8x - 4x 2 a) __________ 4x -3x2 + 5x b) __________ -x 0.6 m For help with #8 and #9, refer to Example 2 on w page 274. a) What polynomial expression represents 8. Divide. the area of the base of the tank? 2y + 4.2y 2 a) __________ b) What polynomial expression represents 2y the length of the tank? 12m - 6.2m + 24 2 b) _________________ c) What is the length of the tank if the 2 width is 0.6 m? What is the volume -18y - 6y 2 of the tank? c) ___________ -6y 3cv - 2.7c 12. For their Valentine’s Day dance, the grade d) __________ 3c 9 students want to decorate the end wall of the gym with red poster paper. The 9. Determine each quotient. area of the wall is given by the polynomial 2.7c + 3.6c 2 45x2 + 20x. One sheet of poster paper a) ___________ covers an area given by the monomial 5x. 3c 2x2 + 8xy What polynomial expression represents the b) _________ number of sheets of paper the students will x need to cover the wall? -s - 1.5st 2 c) ___________ 5s 13. A rectangle has an area of 9x2 - 3x square -14w - 7w + 0.5 2 units. The width of the rectangle is 3x units. d) __________________ 0.5 What is the length? Apply 10. A dump truck holds 10 m3 of soil. You are 9x2 - 3x 3x filling a rectangular space in a yard with the dimensions of (2x + 3) by 5x by 2, in metres. What polynomial expression represents the number of truck loads of soil you will need? 276 Chapter 7 NEL 14. The formula used to predict the distance an 16. Two rectangles have common sides with object falls is d = 4.9t2 + vt. In the formula, a right triangle, as shown. The areas and d is the distance, in metres, t is the time, in widths of the rectangles are shown. What seconds, and v is the starting velocity of the is a simplified expression for the area of object, in metres per second. the triangle? a) The average speed of a falling object is 2x calculated as s = __d , where s is the average t A = 4x2 + 2x speed, in metres per second. Use this information to develop a formula for the average speed of a falling object in terms x of t and v. A = 3x b) What is the average speed of an object that falls for 5 s, if it starts from a resting 17. What is the ratio of the area of the shaded position? rectangle to the area of the large rectangle? Extend x 15. Divide. 3.6gf + 0.93g 2x - 3 x 2x - 3 a) _____________ 0.3g x+6 2 b2 - __ __ 1 ab + __ 1b x 3 3 3 b) ________________ 1b __ 3 18. If a rectangle has length 2xy and area -4.8x + 3.6x - 0.4 2 12x2y + 6xy2, what is its perimeter? c) ___________________ 0.2 You are designing a park that includes a large parking lot that will be covered with gravel. xm a) Design two different-shaped parking lots using any single shape (x + 4) m or combination of regular shapes. Include the dimensions on a 1m drawing of each parking lot design. Note that you will need to be able to calculate the area of your parking lots. Each area should be a different shape. Make them no less than 200 m2 and no greater than 650 m2. b) A truck with dimensions similar to those shown in the picture will deliver the gravel. Write an expression for the approximate area that a single load of gravel will cover to a depth of 5 cm. c) There are three sizes of trucks that can deliver the gravel. The widths are 1.5 m, 2 m, and 3 m. Approximately how many truckloads would it take for each truck size to deliver the required amount of gravel for each of your parking lots? You will cover each parking lot to a depth of 5 cm. Show your work. d) Which truck size do you think would be the most efficient to use for each of your parking lots? Explain your reasoning. NEL 7.3 Dividing Polynomials by Monomials 277  Chapter 7 Review Key Words 10. A square is inscribed in a circle with radius r as shown. What is the ratio of the area of For #1 to #4, match the polynomial in Column A with the square to the area of the circle? an equivalent polynomial in Column B. Polynomials in Column B may be used more than once or not at all. Column A Column B r 8xy A 4xy - 2x 1. ____ 2x B 4x2 - 2x 12x2 - 6x 2. __________ 3x C 4y 3. (-2x + 1)(-2x) D 2x2 - 2x 12xy - 6x 7.2 Multiplying Polynomials by Monomials, 4. __________ E 4xy 3 pages 264–271 F 4x - 2 11. What polynomial multiplication statement is represented by each area model? a) 3y 5 b) 1.2f 7.1 Multiplying and Dividing Monomials, pages 254–263 1.3y f 5. Use a model to complete each monomial multiplication statement. 4 a) (3x)(5x) b) (4x)(-5y) 12. What polynomial multiplication statement is represented by the algebra tiles? 6. Find each product. a) a) (-3.2x)(-2.7y) b) ( __73 a )(-14a) 7. Use a model to complete each monomial division statement. 6x2 a) ____ 2x b) 15a2 ÷ (-3a) 8. Determine each quotient. b) -4.8r a) ______ 2 -1.2r b) 2xy ÷ 2x 9. A rectangle is four times as long as it is wide. If the area of the rectangle is 1600 cm2, what are its dimensions? 278 Chapter 7 NEL 13. Expand. 17. A triangle has an area represented by a) (20x)(2.3x - 1.4) 3x2 + 6x. If the base of the triangle is 3x, what is the height? b) ( __32 p )( p - __43 ) 14. The length of a piece of rectangular A = 3x2 + 6x cardboard in centimetres is 6x + 3. 1 of the length. The width is 1 cm less than __ 3 What is an expression for the area of the 3x cardboard? 18. A rectangular wall has a circular window. The area of the wall can be represented by 32x2 + 16x. The length of the wall is 8x. The diameter of the circular window has a measurement that is half the width 6x + 3 of the wall. What is the radius of the window written as an expression in terms 7.3 Dividing Polynomials by Monomials, of x? pages 272-277 15. Determine the division statement represented by the algebra tiles. Give the quotient. a) ? 8x A = 32x2 + 16x b) ? 19. Naullaq is cutting ice blocks from the lake for her mother's drinking water tank. The cylindrical tank has a volume of 4x2π. Once each block has melted, it will have a volume of 3x2. How many blocks does she need to cut so that her mother’s tank will be filled when the ice 16. Divide. melts? Give your answer to the nearest 12n - 2n2 a) __________ whole block. Explain your answer. 2n 15x - 3x b) __________ 2 1.5x Use the π key on your calculator. NEL Chapter 7 Review 279 Chapter 7 Practice Test For #1 to #6, select the best answer. 4. Which is equivalent to -27q2 ÷ 9q? 1. Which monomial multiplication statement A 3q2 B 3q is represented by the algebra tiles? C -3q D -3q2 2 (3 ) 5. Which is equivalent to __x (-3x - 6)? A -2x - 4x 2 B -2x - 4 C 2x - 4 D 2x2 - 4x 15y - 10y 2 6. Calculate __________. -5y A (3x)(-2x) = -6x2 A -3y - 2 B -3y + 2 B (2x)(-3x) = -6x2 C 3y - 2 D 3y + 2 C (2x)(3x) = 6x2 D (-2x)(-3x) = 6x2 Complete the statements in #7 and #8. -24x + 8xz 2 7. The expression ____________ is equivalent 2. What is the product of 3y and 2.7y? 4x to. A 0.9y B 8.1y C 0.9y2 D 8.1y2 8. A polynomial multiplication expression 2 that is equivalent to 24d - 12d is. 3. Which monomial division statement is represented by the algebra tiles? Short Answer ? For #9 to #11, show all of the steps in your solutions. 9. Calculate (2.4x)(4y). -3 10. What is the product of 12h and ___ h + 2? 4 2x2 + 3x 11. Simplify ________. -3x 12. Paula is building a rectangular patio. It will have a square flower bed in the middle. The rest will have paving stones. The patio will -6x 2 have a length of 4x and a width of 3.1x. The A _____ = -2x -3x area of the flower bed will be 3.5x2. What -6x 2 B _____ = 2x area of the patio will need paving stones? -3x 6x2 C ____ = -2x A = 3.5x2 -3x 6x2 D ____ = 2x 4x -3x 3.1x 280 Chapter 7 NEL 13. A sports field is 15 m longer than twice its Extended Response width. What is an expression for the area of 15. a) What error did Karim make when the field in terms of its width, w? Expand completing the division statement shown? the expression. -18d - 6d 2 _________ 3d 14. The area of a rectangular sandbox can be -18d2 ___ _____ 6d = + expressed as 24xy + 36x. The width of the 3d 3d sandbox is 6x. What is the perimeter of = -6d + 2 the sandbox? b) Show the correct method. A = 24xy + 36x 16. A square with a side length of 2s has a smaller square inscribed. The vertices of the smaller square are at the midpoints of the sides of the larger square. What is the ratio of the area of the larger square to the area of the smaller square? Express your answer in its simplest form. 6x 2s You have been hired to create a landscape design for a park. The park is rectangular and covers an area of 500 000 m2. The park includes the following features: a play area covered with bark mulch a sand area for playing beach volleyball a wading pool The features in your design include the following shapes: a circular area a rectangular area a parallelogram-shaped area with the base three times the height The features of your park have varying depths. Include the following in your design: a scale drawing showing the layout of each of the required features a list showing the area of each feature and the volume of each material (mulch, sand, and water) required to complete the park a polynomial expression for the area and volume of each feature, using a variable for one of the dimensions NEL Chapter 7 Practice Test 281  Challenges Design a Card Game Materials at least 30 index cards You are a game designer. You are going to create a skill-testing card game per pair of students , or that involves the multiplication and division of polynomial expressions. heavy paper for cutting out cards Work with a partner. You will create at least 15 pairs of cards, using the following guidelines: For each pair, on one card write a polynomial expression that involves One card might be multiplication or division. On the other card write an equivalent (5x)(4x), and the expression. equivalent card might Include the types of polynomial expressions found in each section of be 20x2. Chapter 7. Use a variety of constants, monomials, and polynomials. 1. Create a list of the pairs of expressions for your game. 2. Create the cards. 3. Write the rules for your game. Your game should be suitable for two or more players. 4. Exchange your cards and rules with another set of partners. Play the game. 282 Chapter 7 NEL Polynomial Puzzle Materials sample polynomial 1. Try the nine-piece puzzle below. puzzle per student -3x + 2 3x - 4 scissors z) 9xz ÷ (-3x (4x)(-5y) 24x2 15 2 __x__ - __10 __x_ 5x(3 - 2y) -5x -5y (2x + 3y) x+3 -4.8x 2 ÷ 1.2x 2xy ÷ 2x -20xy 12yz y 2 - 12x 9x ________ 2x(3x - 5) 3x 15x - 10xy -10xy - 15 6x2 - 10x y2 (-3x)(-8x) (3yz)(4) -4x -3 3xy + 9y ________ 3y a) Cut out the pieces of a copy of the puzzle. b) Solve the puzzle. Polynomial expressions involving multiplication or division are red. Matching equivalent expressions are blue. Match each multiplication or division expression with its equivalent expression. The diagram shows how a solved nine-piece puzzle would look. 2. Design your own 16-piece puzzle. a) Draw 16 equal-sized squares on a piece of paper. b) Write your own matching expressions. Place them the same way as in the small diagram above so that each expression and its equivalent are across from each other. Include the types of polynomial expressions found in Chapter 7. Include a variety of constants, monomials, and polynomials. Cover a variety of difficulty levels. c) Cut out your sixteen puzzle pieces. Mix them up. d) Exchange your puzzle with a classmate’s. Solve your classmate’s puzzle. NEL Challenges 283  Chapters 5-7 Review Chapter 5 Introduction to Polynomials a) Write an algebraic expression for the 1. The following diagram of algebra tiles antique shop’s total income from the models a backyard. sale of comics, hardcover books, and paperbacks. Tell what each variable represents. b) Use your expression to show the total income after the sale of 15 comics, 7 hardcover books, and 5 paperbacks. a) What is an expression for the perimeter c) One day, the store sold $100 worth of of the backyard? comics, hardcovers, and paperbacks. b) What is an expression for the area of the What number of each item did the store backyard? sell? Show that more than one answer is possible. 2. Use materials or diagrams to show the collection of like terms in each expression. 6. A park is divided equally into three square a) 8c + 3 - 5c + 1 sections. Each section will have a side measurement of 2n + 4. The park will have b) -1 + x - 1 - x + 1 fencing built as shown. Each opening has c) g2 - g + 5 + 2g - 4g2 length n and does not need any fencing. What is the total length of fencing needed 3. Write the following expressions in to complete the job? simplest form. a) (2m - 3) + (5m + 1) b) (w2 - 4w + 7) + (3w2 + 5w - 3) c) (9y2 - 6.8) + (4.3 - 9y - 2y2) 4. Write a simpler expression. a) (-7z + 3) - (-4z + 5) Chapter 6 Linear Relations 7. a) Describe the relationship between the b) (3d - 2d - 7) - (d 2 - 5d + 6cd - 2) figure number and the number of tiles. c) (2x2 + 3xy) - (-xy + 4x2) 5. The Better Buys antique shop sells comic books for $10, hardcover books for $8, and paperback novels for $3. Figure 1 Figure 2 Figure 3 $8 $3 b) Develop a linear equation to model the pattern. c) If the pattern were continued, how many $10 squares would be in Figure 8? 284 Chapter 7 NEL  8. Monika is saving money for a ski trip. She 10. A computer salesperson earns a monthly starts with $112 in her bank account. She salary plus a 10% commission on each sale. decides to deposit $25 every week until she The sales, commission, and earnings are has enough money to pay for the trip. shown in the table. a) Create a table of values for the first Sales ($) Commission ($) Earnings ($) five deposits. 0 0 2000 b) What linear equation models 5 000 500 2500 this situation? 10 000 1000 3000 c) If Monika needs at 15 000 1500 3500 least $450 for her trip, for 20 000 2000 4000 how many 25 000 2500 4500 months a) Draw a graph showing the linear relation does she between sales and earnings. need to b) The salesperson earns $3750 in one deposit money into her bank account? month. What are the approximate total sales for that month? 9. A car mechanic charges a $35 base fee for c) The salesperson earns $5500 the next labour plus an hourly rate of $60. The month. What are the approximate total graph shows this linear relation. sales for that month? C Repair Costs d) Approximately how much would the 350 salesperson have to sell in order to earn 300 $4250 in one month? Cost ($) 250 200 11. Draw the graph that represents this table 150 100 of values. 50 Time (h) 0.25 0.50 0.75 1.00 1.25 1.50 1.75 0 1 2 3 4 5 6 7 8 t Cost ($) 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Time (h) a) Describe a situation to represent the data a) Approximately how much would the on the graph. mechanic charge after working on a b) Write an equation to model the data. vehicle for 8 h? c) What is the cost for 3.25 h? b) Approximately how many hours would a mechanic work to charge $225 in labour costs? c) Another mechanic charges at the same rate, but in half-hour increments. What would be the cost of a repair that took 9.5 h to complete? NEL Chapters 5–7 Review 285 Chapter 7 Multiplying and Dividing 16. Determine each quotient. Polynomials 12g + 8g 2 a) _________ 12. Find the product of each pair of monomials. 4g a) (3x)(4x) -6x + 3xy 2 b) ___________ b) (2.5y)(-4y) 3x c) (s)(-0.5s) c) (9.3ef 2 - 62e) ÷ (-3.1e) t ( ) d) __ (10t) 5 d) (24n2 + 8n) ÷ (0.5n) 17. A rectangle has an area represented 13. Divide. by the expression 10x2 - 5x. If the a) 8.4x ÷ x 2 length of the rectangle is 5x, what is b) (-12h2) ÷ 2h an expression for the width, w, of c) (-0.6n2) ÷ (-0.2n) the rectangle in terms of x? 4.8p 2 d) ______ -1.2p w 14. Use an area model to expand each expression. a) (3x)(2x + 1) 5x b) (5w + 3)(1.5w) 15. If a foosball table is 3 cm longer than twice its width, what is an expression for the area of the foosball table? Express your answer in expanded form. Digital rights not available. 286 Chapter 7 NEL  Task Choosing a Television to Suit Your Room You want to find a television that Materials best suits your needs, and measuring tape considers your room size and the location for the television. grid paper Does a standard or high-definition television (HDTV) make the most sense for your room? How large of a screen should you get? 1. The following table gives you the best viewing distance for the screen size for two types of TVs. Screen Size Viewing Distance (cm) (cm) Standard TV HDTV 68.8 205.7 172.7 81.3 243.8 203.2 94.0 281.9 233.7 a) Given this information, what size of television would be best for your classroom? Make a sketch of your classroom, including where you plan to place the TV and the best place for a student to view it from. b) If the television is 320 cm away from your seat, how large of a standard TV would be best? c) How will your answer for part b) change if you have a HDTV? 2. The diagram shows the viewing angles for various types of televisions. Calculate the viewing area of the standard TV type and size of plasma LCD your choice. 3. What type and size of TV would be best Viewing angle: for a room in your standard: 120° home? Justify your plasma: 160° response. LCD: 170° NEL Task 287