ACTIVITY: Multiplying Binomials Using Algebra Tiles. Work with a partner. Six different algebra tiles are shown below.

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1 7.3 Multiplying Polynomials How can you multiply two binomials? 1 ACTIVITY: Multiplying Binomials Using Algebra Tiles Work with a partner. Six different algebra tiles are shown below. 1 1 x x x x Write the product of the two binomials shown by the algebra tiles. a. (x + 3)(x ) = b. (x 1)(x + 1) = c. (x + )(x 1) = d. ( x )(x 3) = COMMON CORE Polynomials In this lesson, you will multiply binomials using the Distributive Property, a table, or the FOIL method. multiply binomials and trinomials. Learning Standard A.APR Chapter 7 Polynomial Equations and Factoring

2 Math Practice Use a Diagram How can you represent the product of polynomials using diagrams? ACTIVITY: Multiplying Monomials Using Algebra Tiles Work with a partner. Write each product. Explain your reasoning. a. b. c. d. e. f. g. h. i. j. 3 ACTIVITY: Multiplying Binomials Using Algebra Tiles Use algebra tiles to find each product. a. (x )(x + 1) b. (4x + 3)(x ) c. ( x + )(x + ) d. (x 3)(x + 4) e. (3x + )( x 1) f. (x + 1)( 3x + ) g. (x ) h. (x 3) 4. IN YOUR OWN WORDS How can you multiply two binomials? Use the results of Activity 3 to summarize a procedure for multiplying binomials without using algebra tiles. 5. Find two binomials with the given product. a. x 3x + b. x 4x + 4 Use what you learned about multiplying binomials to complete Exercises 3 and 4 on page 345. Section 7.3 Multiplying Polynomials 341

3 7.3 Lesson Lesson Tutorials Key Vocabulary FOIL Method, p. 343 In Section 1., you used the Distributive Property to multiply a binomial by a monomial. You can also use the Distributive Property to multiply two binomials. EXAMPLE 1 Multiplying Binomials Using the Distributive Property Find each product. a. (x + )(x + 5) Use the horizontal method. Distribute (x + 5) to (x + )(x + 5) = x(x + 5) + (x + 5) each term of (x + ). = x(x) + x(5) + (x) + (5) Distributive Property = x + 5x + x + 10 Multiply. = x + 7x + 10 Combine like terms. b. (x + 3)(x 4) Use the vertical method. Multiply 4(x + 3). x + 3 x 4 Align like terms vertically. 4x 1 Distributive Property Multiply x(x + 3). x + 3x Distributive Property x x 1 Combine like terms. The product is x x 1. EXAMPLE Multiplying Binomials Using a Table Find (x 3)(x + 5). Step 1: Write each binomial as a sum of terms. (x 3)(x + 5) = [x + ( 3)](x + 5) Step : Make a table of products. The product is x 3x + 10x 15, or x + 7x 15. x 3 x x 3x 5 10x 15 Exercises 5 13 and 16 1 Use the Distributive Property to find the product. 1. ( y + 4)( y + 1). (z )(z + 6) Use a table to find the product. 3. ( p + 3)( p 8) 4. (r 5)(r 1) 34 Chapter 7 Polynomial Equations and Factoring

4 The FOIL Method is a shortcut for multiplying two binomials. FOIL Method To multiply two binomials using the FOIL Method, find the sum of the products of the First terms, (x + 1)(x + ) x(x) = x Outer terms, (x + 1)(x + ) x() = x Inner terms, and (x + 1)(x + ) 1(x) = x Last terms. (x + 1)(x + ) 1() = (x + 1)(x + ) = x + x + x + = x + 3x + EXAMPLE 3 Multiplying Binomials Using the FOIL Method Find each product. a. (x 3)(x 6) First Outer Inner Last (x 3)(x 6) = x(x) + x( 6) + ( 3)(x) + ( 3)( 6) Use the FOIL Method. = x + ( 6x) + ( 3x) + 18 Multiply. = x 9x + 18 Combine like terms. b. (x + 1)(3x 5) First Outer Inner Last (x + 1)(3x 5) = x(3x) + x( 5) + 1(3x) + 1( 5) Use the FOIL Method. = 6x + ( 10x) + 3x + ( 5) Multiply. = 6x 7x 5 Combine like terms. Exercises 30 Use the FOIL Method to find the product. 5. ( m + 5)( m 6) 6. (x 4)(x + ) 7. (k + 5)(6k + 3) 8. ( u + 1 ) ( u 3 ) Section 7.3 Multiplying Polynomials 343

5 EXAMPLE 4 Multiplying a Binomial and a Trinomial Find (x + 5)(x 3x ). x 3x x + 5 Align like terms vertically. Multiply 5(x 3x ). 5x 15x 10 Distributive Property Multiply x(x 3x ). x 3 3x x x 3 + x 17x 10 The product is x 3 + x 17x 10. Distributive Property Combine like terms. EXAMPLE x ft (x 7) ft (x + 10) ft 5 Real-Life Application In hockey, a goalie behind the goal line can only play a puck in a trapezoidal region. a. Write a polynomial that represents the area of the trapezoidal region. 1 h(b 1 + b ) = 1 (x 7)[x + (x + 10)] Substitute. = 1 (x 7)(x + 10) Combine like terms. F O I L = 1 [x + 10x + ( 14x) + ( 70)] Use the FOIL Method. = 1 (x 4x 70) Combine like terms. = x x 35 Distributive Property b. Find the area of the trapezoidal region when the shorter base is 18 feet. Find the value of x x 35 when x = 18. x x 35 = 18 (18) 35 Substitute 18 for x. = Simplify. = 53 Subtract. The area of the trapezoidal region is 53 square feet. Exercises Find the product. 9. ( x + 1)( x + 5x + 8) 10. (n 3)(n n + 4) 11. WHAT IF? How does the polynomial in Example 5 change if the longer base is extended by 1 foot? Explain. 344 Chapter 7 Polynomial Equations and Factoring

6 7.3 Exercises Help with Homework 1. VOCABULARY Describe two ways to find the product of two binomials.. WRITING Explain how the letters of the word FOIL can help you remember how to multiply two binomials. 9+(-6)=3 3+(-3)= 4+(-9)= 9+(-1)= Write the product of the two binomials shown by the algebra tiles. 3. (x )(x + ) = 4. ( x + 3)(x 1) = 1 Use the Distributive Property to find the product. 5. (x + 1)(x + 3) 6. ( y + 6)( y + 4) 7. (z 5)(z + 3) 8. (a + 8)(a 3) 9. (g 7)(g ) 10. (n 6)(n 4) 11. (3m + 1)(m + 9) 1. (p 4)(3p + ) 13. (6 5s)( s) 14. ERROR ANALYSIS Describe and correct the (t )(t + 5) = t (t + 5) error in finding the product. = t t 10 = t CALCULATOR The width of a calculator can be represented by (3x + 1) inches. The length of the calculator is twice the width. Write a polynomial that represents the area of the calculator. Section 7.3 Multiplying Polynomials 345

7 Use a table to find the product. 16. (x + 3)(x + 1) 17. ( y + 10)( y 5) 18. (h 8)(h 9) 19. ( 3 + j )(4 j 7) 0. (5c + 6)(6c + 5) 1. (5d 1)( 7 + 3d) 3 Use the FOIL Method to find the product.. (b + 3)(b + 7) 3. (w + 9)(w + 6) 4. (k + 5)(k 1) 5. (x 4)(x + 8) 6. (q 3)(q 4) 7. (z 5)(z 9) 8. (t + )(t + 1) 9. (5v 3)(v + 4) 30. (9 r)( 3r) 31. ERROR ANALYSIS Describe and correct the error in finding the product. (r + 6)(r 7 ) = r (r) + r (7) + 6(r ) + 6(7) = r + 7r + 6r + 4 = r + 13r OPEN-ENDED Write two binomials whose product includes the term 1. (x + 10) yd 33. SOCCER The soccer field is rectangular. (x 30) yd a. Write a polynomial that represents the area of the soccer field. b. Use the polynomial in part (a) to find the area of the field when x = 90. c. A groundskeeper mows 00 square yards in 3 minutes. How long does it take the groundskeeper to mow the field? Write a polynomial that represents the area of the shaded region y 3 p 6 x 5 x 1 y p + 1 x 4 x + Find the product. 37. (n + 3)(n + 1) 38. (x + y )(x y ) 39. (r + s)(r 3s) (x 4)(x 3x + ) 41. (f + 4f 8)(f 1) 4. (3 + i )(i + 8i ) 43. (t 5t + 1)( 3 + t) 44. (b 4)(5b 5b + 4) 45. (3e 5e + 7)(6e + 1) 46. REASONING Can you use the FOIL method to multiply a binomial by a trinomial? a trinomial by a trinomial? Explain your reasoning. 346 Chapter 7 Polynomial Equations and Factoring

8 47. AMUSEMENT PARK You go to an amusement park (x + 1) times each year and pay (x + 40) dollars each time, where x is the number of years after 011. a. Write a polynomial that represents your yearly admission cost. b. What is your yearly admission cost in 013? 48. PRECISION You use the Distributive Property to multiply (x + 3)(x 5). Your friend uses the FOIL Method to multiply (x 5)(x + 3). Should your answers be equivalent? Justify your answer. 49. REASONING The product of (x + m)(x + n) is x + bx + c. a. What do you know about m and n when c < 0? b. What do you know about m and n when c > 0? 50. PICTURE You design the wooden picture frame and paint the front surface. x in. (x 3) in. (x 4) in. a. Write a polynomial that represents the area of wood you paint. b. You design the picture frame to display a 5-inch by 8-inch photograph. How much wood do you paint? (x 7) in. 51. The shipping container is a rectangular prism. Write a polynomial that represents the volume of the container. (x + ) ft (4x 3) ft (x + 1) ft Write the polynomial in standard form. Identify the degree and classify the polynomial by the number of terms. (Section 7.1) 5. x 5x x z 5 7 z y MULTIPLE CHOICE Which system of linear equations does the graph represent? (Section 4.1) A y = 3x + 4 B y = x + 1 y = x 6 y = x C y = x + 7 D y = x + 10 y = 4x 8 y = 3x + y x Section 7.3 Multiplying Polynomials 347

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