Lesson 10.4 Multiplying Binomials

Transcription

1 Lesson 0.4 Multiplying Binomials Objectives Use algebra tiles and geometric representations to multiply two binomials. Use the FOIL method to multiply two binomials. A small patio is 4 feet by 6 feet. You want to epand the patio by the same length on each side. What formula epresses the new area? 4 feet 6 feet Eamine the diagram of the patio. Let represent the amount added to each side of the patio. Then one side will be represented by 4 and the other side by 6. The epression for the new area of the patio will be the product of the binomials. ( 4)( 6) 6 4 Notice you can divide the new area into four regions with the following areas Multiplying Binomials 577

2 The formula for total area is the sum of the areas or Suppose you need to epand the patio by 3 feet on each side. Substitute 3 for to check if ( 4)( 6) (3 4)(3 6) 3 2 0(3) 24 (7)(9) Eample Multiplying Two Binomials Multiply ( 2)( ). Solution The product ( 2)( ) can represent a rectangle with length ( 2) and width ( ). The area of this rectangle is shown below Area of rectangle equals ( + 2)( + ) If you label the area for each of the small rectangles, you can determine the total area of the overall rectangle. This area is the same as the product of ( 2)( ) Adding separate areas: Area = Area = You have just used a geometric representation to multiply two binomials, 2 and. The result is the trinomial Thus, ( 2)( ) Chapter 0 Polynomials and Factors

3 The geometric model can be used to derive an algebraic pattern for multiplying two binomials. Consider the product ( 3)(2 ). Let ( 3) be the length of a rectangle and (2 ) be the width The development of the algebraic product ( 3)(2 ), term by term, is shown in a series of four drawings. The geometric equivalent of each term-byterm product is highlighted by the shaded portions in the rectangle. Step : ( + 3)(2 + ) ()(2) = 2 2 Step 2: ( + 3)(2 + ) ()() = Step 3: ( + 3)(2 + ) (3)(2) = 6 Step 4: ( + 3)(2 + ) (3)() = Multiplying Binomials 579

4 Add together all the highlighted areas in the four rectangles to find the total area of the rectangle of size ( 3) by (2 ) ( 3)(2 ) When the four steps in the multiplication process are done in sequence, the result is the following: ( 3)(2 ) ( 3)(2 ) A simple memory device will help you keep track of which terms are being multiplied. This device is known as the FOIL technique. To find the product of ( + 3)(2 + ), proceed as follows: First terms are multiplied ( 3)(2 ) Outer terms are multiplied ( 3)(2 ) Inner terms are multiplied ( 3)(2 ) Last terms are multiplied ( 3)(2 ) 3 3 Add and then simplify by combining similar terms. ( 3)(2 ) The FOIL method is a shortcut for using the Distributive Property. It helps you multiply two binomials in an orderly manner. Practice will help make the process automatic. 580 Chapter 0 Polynomials and Factors

5 When one or both binomials in a product contains a negative term, the geometric model becomes complicated to use. For eample, if you are asked to find the product ( 2)( ), it is hard to imagine how to represent 2. However, the FOIL method readily handles the negative terms. F O I L ( 2)( ) ()() ()() ( 2)() ( 2)() 2 2 Algebra tiles are another way to use areas to model the multiplication of binomials. Algebra tiles solve the problem of negative areas by using a quadrant system to identify areas as either positive or negative. Eample 2 Using Algebra Tiles to Multiply Binomials Multiply ( 2)( ) using algebra tiles as a model. Solution. Select the tiles that model 2 and. 2. Arrange these tiles along perpendicular lines that represent the length and width of an area model. 3. Now use additional algebra tiles to form a rectangle using the tiles on the aes to model the length and width of the rectangle. Note that the unit tiles placed right of the positive unit tile and above the negative unit tile are negative because a positive multiplied by a negative is a negative. List the si areas that make up the rectangle you formed. 2,,,,, 4. Add the areas of the rectangle formed by the algebra tiles. Do not include in the sum the areas of the algebra tiles outside the lines. What is the sum of the areas? What is the product of ( 2) and ( )? Multiplying Binomials 58

6 Critical Thinking Eplain how the Distributive Property is used to find the following product: ( 3)(2 ) ( 3)(2) ( 3)() 3 is distributed 5 ()(2) (3)(2) ()() (3)() 2 is distributed; is distributed (6 ) 3 is factored out Lesson Assessment Think and Discuss n. How does the geometric model illustrate the product of two binomials? Show an eample. 2. How do you use the FOIL method to find the product of two binomials? Show an eample. 3. Try doing the multiplication by placing one binomial under the other and multiplying as if they were whole numbers. How is this method similar to numerical multiplication? 4. Use the FOIL method to find the product (0 4)(0 6). 5. Eplain how the FOIL method can be used to find the product of two numbers such as (24)(35). Practice and Problem Solving Find the products using a geometric model. 6. ( 4)( 3) ( 5)( ) ( )( 3) 0. ( 4)( 2) 9. ( 2 )( 3) Each pair of factors 0. ( 2 4)( 2 2) represents the lengt 8. (2 )( 2) ( ( 2 3)( ) th and width of a Each pair of factors represents the length and width of a rectangle. Draw an area model and find the area of the rectangle. 2. ( 3)( 3) ( 5)( 2) (3 4)(2 3) (2 )( 6) 6. ( )(2 3) 7. ( )( 5) 5. (2 )( 6) Find each product u 6. ( )(2 3) using the FOIL metho 7. ( )( 5) od.

7 Find each product using the FOIL method. 8. (3 )( 3) 9. (2 3)(4 ) (3 2 5)(2 ) 22. (5 2 2)(3 2 ) (3 6)(2 2 6) 25. (5 2 4)(2 4) (4 2 )(3 2 3) 28. (3 8)(7 2 4) Mied Review 20. (5 )(2 3) 23) (2 2)(3 2 ) (2 8)(3 2 4) (2 29)(6 2 2) 2) Solve each equation for % $65 $50 Write and solve an equation for each situation. 33. Taj and Ramen invested $360 in stock. Taj invested $20 more than Ramen. How much did Taj invest? y 360 and 20 y; $ If a rectangle has a length that is 3 cm less than four times its width, and its perimeter is 9 cm, what are its dimensions? 2L 2W 9 and 4W 3 L; width 2.5 cm; length 7 cm 35. John has $5,000 to invest. He invests part of the money in an investment that pays 2% simple interest and the rest in another that pays 8% simple interest. If his annual income from the two investments is $,456, how much did he invest at 2%? 5,000 y and y,456; $6, Multiplying Binomials 583