Multiplying Polynomials by Monomials A.APR.1

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1 ? LESSON 14.3 ESSENTIAL QUESTION Multiplying Polynomials by Monomials How can you multiply polynomials by monomials? A.APR.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Also A.SSE.2, A.CED.1 EXPLORE ACTIVITY A.APR.1 Modeling Polynomial Multiplication You can use algebra tiles to model the multiplication of a polynomial by a monomial. Rules 1. The first factor goes on the left side of the grid, the second factor on the top. 2. Fill in the grid with tiles that have the same height as tiles on the left and the same length as tiles on the top. 3. Follow the key on the right. The product of two tiles of the same color is positive; the product of two tiles of different colors is negative. A Use algebra tiles to find 2(x 1). STEP 1 Fill in the factors. STEP 2 Fill in the grid. Place the factor 2 on the left. Place the factor x 1 on the top. Fill in the grid according to Rule 2 above. Draw the missing tiles. STEP 3 Count the positive and negative tiles in the grid. KEY = positive variable = negative variable = 1 = 1 x tiles: 1 tiles: Expression: Lesson

2 EXPLORE ACTIVITY (cont d) B Use algebra tiles to find 2x(x - 3). STEP 1 Fill in the factors. Remember, the first factor goes on the left side and the second factor goes on the top row. STEP 2 Fill in the grid. Fill in the grid according to Rule 2. Draw the tiles. Positives are yellow, negatives are red. STEP 3 Count the positive and negative tiles in the grid. x 2 tiles: x tiles: 1 tiles: expression: REFLECT 1. How do the tiles illustrate the idea of x 2 geometrically? 2. How does the grid illustrate the Distributive Property? Explain. 502 Unit 4

3 Multiplying Monomials When multiplying monomials, you may have to multiply variables with exponents. Recall the Product of Powers Property, which states that a m a n = a (mn). EXAMPLE 1 Multiply. A (2 x 4 )(-3 x 5 ) (2-3)( x 4 x 5 ) Group factors that use the same variable. (2-3)( x 45 ) Product of Powers Property -6 x 9 Simplify. B (8 g 2 h 5 )(6 gh 3 ) (8 6)( g 2 g)( h 5 h 3 ) Group factors that use the same variable. A.APR.1 Math On the Spot (8 6)( g 21 )( h 53 ) Product of Powers Property 48 g 3 h 8 Simplify. REFLECT 3. What can you conclude about the order of factors in multiplication? Math Talk Mathematical Practices Is the product of two monomials always a monomial? Explain. 4. Explain the Error Felicity reasons that g 2 g 0.5 = g 1, since = 1. Explain her error and find the correct product for g 2 g Communicate Mathematical Ideas If x 8 x y = x, what is the value of y? Explain. YOUR TURN Find the products x 5 (5 x 4 ) 7. 6 a 4 b(4 a 3 b 2 ) 8. (6 a 2 b 5 )(3ab) 9. (4 k 2 d 5 )(9 d 3 j 2 ) Personal Math Trainer Online Practice and Help Lesson

4 Multiplying a Polynomial by a Monomial Remember, the Distributive Property states that multiplying a term by a sum is the same thing as multiplying the term by each part of the sum, then adding the results. Math On the Spot EXAMPLE 2 Find each product. A.APR.1 My Notes A B 4(5 x 2 3x 1) 4(5 x 2 3x 1) Distribute 4. 4(5 x 2 ) 4(3x) 4(1) Regroup and multiply. 20 x 2 12x 4 Simplify. 2x (3 x 2 2x - 4) 2x(3 x 2 2x - 4) Distribute 2x. Remember, 2x = 2 x 1. 2 x 1 (3 x 2 ) 2 x 1 (2 x 1 ) - 2 x 1 (4) Regroup. 6 x 12 4 x 11-8 x 1 Multiply and combine exponents. 6 x 3 4 x 2-8x Simplify. REFLECT 10. Is the product of a monomial and a polynomial always a polynomial? Explain. If so, how many terms does it have? YOUR TURN Find each product. 11. (4 m 3 n 2 )(5 m 2 n - 3mn - 2) 12. 2a 2 ( 5b 2 3ab 6a 1) Personal Math Trainer Online Practice and Help 13. 3ab(2 a 2 b 6ab 2 8b) 504 Unit 4

5 Application of Multiplying Polynomials and Monomials Knowing how to multiply polynomials and monomials is useful when solving real-world problems. EXAMPLE 3 A.APR.1, A.CED.1 Math On the Spot Chrystelle is making a planter box with a square base. She wants the height of the box to be 3 inches more than the length. If she needs the volume of the box to be as close as possible to 6,000 i n 3, what should the length of the box be to the nearest whole inch? STEP 1 Define terms and write what you know. side length = s height = s 3 volume = (side length) (side length) height = (s s)(s 3) = s 2 (s 3) = s 3 3 s 2 Image Credits: intrepidina/shutterstock STEP 2 STEP 3 Make a plan. Chrystelle wants the volume of the planter to be as close as possible to 6000 i n 3, so you need to find a value for s that gives a product that is close to 6000 i n 3. Select the best answer. 6,000 is closer to 5780 than it is to 6804, so the length of the planter to the nearest whole inch should be 17 inches. REFLECT 14. What If? Explain how the answer would change if Chrystelle wanted the volume as close as possible to 4,400 in 3. YOUR TURN ss 3 3 s (15) 2 = (16) 2 = (17 )2 = (18) 2 = David needs a piece of paper where the length is 4 inches more than the width, and the area is as close as possible to 50 in 2. To the nearest whole inch, what should the measurements be for the piece of paper? Personal Math Trainer Online Practice and Help Lesson

6 Guided Practice Use the algebra tiles to find the product of polynomials. (Explore Activity) 1. What multiplication problem is being modeled? 2. Fill in the grid. _ 3. What is the product? Find the product of the monomials. (Example 1) 4. d 5 f 2 (16 d 3 e 4 f 2 ) = ( ) ( ) r 3 s 5 (-3rs 2 ) = = = Find the product of each monomial and polynomial. (Example 2) 6. 5(2 k 2 k 3) 5( ) 5( ) 5( ) 7. 8t(3 t 2 5ts 2s) ( ) ( ) 8t( ) 8. A jeweler sells gems at a cost in dollars per centigram that is 4 more than 6 times the weight of the gem in centigrams. Thus, a gem that weighs w centigrams will be sold at a price of 6w 4 dollars per centigram meaning that a larger, heavier gem will cost more per centigram than a smaller gem. If the jeweler sold a single gem for about $2500, what did the gem weigh to the nearest centigram? (Example 3)? ESSENTIAL QUESTION CHECK-IN 9. How can you multiply a polynomial by a monomial? 506 Unit 4

7 Name Class Date 14.3 Independent Practice A.APR.1, A.SSE.2, A.CED.1 Personal Math Trainer Online Practice and Help Find each product x(3 x 3 y 4 ) m(-16 m 4 n 2 ) a 2 b 4 (-7 a 3 b 2 ) 13. 5(2 k 2 k 4) 14. 9j(6 k 2-2k 13j) a 2 b 2 (-4 a 3 b 5 ) 16. 3i(3 i 2 3ig 3 i 2 g 2 ) t(3 t 3 7ts 5s) de 2 (-4 d 2 12de 2-8 d 2 e) v 2 w 3 ( v 2-11 v 2 w 5 6 w 4 ) a 3 b 4 (-3 a 2 8ab 2-7 a 2 b) x 4 y 4 ( x 4-4 x 4 y 4 4 y 3 ) Image Credits: D. Hurst/Alamy 22. Interpret the Answer A construction engineer needs to make 25 square concrete slabs with sides of length x feet and height x - 3 feet. If the engineer can use at most 350,000 f t 3 of concrete, what should the dimensions of each slab be, to the nearest foot? a. Write an expression for the volume of all the slabs. b. What should x equal, to the nearest foot? 23. A fish market sells sushi-grade fish at a price of 4f - 2 dollars per foot, meaning that the larger the fish is, the greater its price per foot will be. If the market sold a fish for $552, what was the length of the fish? x - 3 x x Lesson

8 24. Analyze Relationships The target heart rate for a fit person of age a exercising at p percent of his or her heart rate is determined by the expression 1_ 1_ 2 p(418 - a) for women and 2 p(400 - a) for men. What is the difference between the target heart rates for a fit woman and a fit man? FOCUS ON HIGHER ORDER THINKING Work Area 25. Represent Real World Problems The Reflecting Pool near the Washington Monument is a rectangle with a length that is 50 feet more than 15 times its width. The depth of the pool is 1 50 of the width. Let w represent the width of the pool. a. Write polynomial expressions for the length and depth of the pool in terms of w. Then write a polynomial expression for the volume of the pool in terms of w. b. The depth of the pool is 3 feet. What is the volume of the pool? 26. Explain the Error Sandy says the product of x 2 and x 3 5 x 2 1 is x 6 5 x 4 x 2. Explain the error that Sandy made and give the correct product. 27. Critical Thinking You are finding the product of a monomial and a binomial. How is the degree of the product related to the degree of the monomial and the degree of the binomial? Give examples and explain your reasoning. 508 Unit 4