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Four Methods to Multiply Binomials

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1 Four Methods to Multiply Binomials
(x + 2)(x + 5) Becky Afghani Long Beach Unified School District 2006

2 What are the 4 methods?

3 What are the 4 methods? The Vertical Method

4 What are the 4 methods? The Vertical Method FOIL

5 What are the 4 methods? The Vertical Method FOIL Algebra Tiles

6 What are the 4 methods? The Vertical Method FOIL Algebra Tiles The Generic Rectangle

7 Method 1: Vertical Method
(x + 2) (x + 5)

8 Method 1: Vertical Method
(x + 2) (x + 5) 1. Line up vertically to multiply.

9 Method 1: Vertical Method
(x + 2) (x + 5) (x + 2) (x + 5) 1. Line up vertically to multiply.

10 Method 1: Vertical Method
(x + 2) (x + 5) (x + 2) (x + 5) 1. Line up vertically to multiply. 2. Multiply the 5 from the bottom by each of the addends in the top.

11 Method 1: Vertical Method
(x + 2) (x + 5) (x + 2) (x + 5) 1. Line up vertically to multiply. 2. Multiply the 5 from the bottom by each of the addends in the top. + 10

12 Method 1: Vertical Method
(x + 2) (x + 5) (x + 2) (x + 5) 1. Line up vertically to multiply. 2. Multiply the 5 from the bottom by each of the addends in the top. 5x + 10

13 Method 1: Vertical Method
(x + 2) (x + 5) (x + 2) (x + 5) 1. Line up vertically to multiply. 2. Multiply the 5 from the bottom by each of the addends in the top. 5x + 10 3. Multiply the x from the bottom by each of the addends in the top. Line up like terms.

14 Method 1: Vertical Method
(x + 2) (x + 5) (x + 2) (x + 5) 1. Line up vertically to multiply. 2. Multiply the 5 from the bottom by each of the addends in the top. 5x + 10 + 2x 3. Multiply the x from the bottom by each of the addends in the top. Line up like terms.

15 Method 1: Vertical Method
(x + 2) (x + 5) (x + 2) (x + 5) 1. Line up vertically to multiply. 2. Multiply the 5 from the bottom by each of the addends in the top. 5x + 10 x2 + 2x 3. Multiply the x from the bottom by each of the addends in the top. Line up like terms.

16 Method 1: Vertical Method
(x + 2) (x + 5) (x + 2) (x + 5) 1. Line up vertically to multiply. 2. Multiply the 5 from the bottom by each of the addends in the top. 5x + 10 x2 + 2x 3. Multiply the x from the bottom by each of the addends in the top. Line up like terms. 4. Add the like terms

17 Method 1: Vertical Method
(x + 2) (x + 5) (x + 2) (x + 5) 1. Line up vertically to multiply. 2. Multiply the 5 from the bottom by each of the addends in the top. 5x + 10 x2 + 2x 3. Multiply the x from the bottom by each of the addends in the top. Line up like terms. 4. Add the like terms

18 Method 1: Vertical Method
(x + 2) (x + 5) (x + 2) (x + 5) 1. Line up vertically to multiply. 2. Multiply the 5 from the bottom by each of the addends in the top. 5x + 10 x2 + 2x 3. Multiply the x from the bottom by each of the addends in the top. Line up like terms. + 10 4. Add the like terms

19 Method 1: Vertical Method
(x + 2) (x + 5) (x + 2) (x + 5) 1. Line up vertically to multiply. 2. Multiply the 5 from the bottom by each of the addends in the top. 5x + 10 x2 + 2x 3. Multiply the x from the bottom by each of the addends in the top. Line up like terms. + 7x + 10 4. Add the like terms

20 Method 1: Vertical Method
(x + 2) (x + 5) (x + 2) (x + 5) 1. Line up vertically to multiply. 2. Multiply the 5 from the bottom by each of the addends in the top. 5x + 10 x2 + 2x 3. Multiply the x from the bottom by each of the addends in the top. Line up like terms. x2 + 7x + 10 4. Add the like terms

21 Method 1: Vertical Method
(x + 2) (x + 5) (x + 2) (x + 5) 1. Line up vertically to multiply. 2. Multiply the 5 from the bottom by each of the addends in the top. 5x + 10 x2 + 2x 3. Multiply the x from the bottom by each of the addends in the top. Line up like terms. x2 + 7x + 10 4. Add the like terms

22 Method 2: FOIL (x + 2) (x + 5)

23 Method 2: FOIL (x + 2) (x + 5) 1. Multiply the First terms
Write this down! 2. Then the Outer terms 3. Then the Inner terms 4. Then the Last terms

24 Method 2: FOIL (x + 2) (x + 5) F O I L 1. Multiply the First terms
2. Then the Outer terms I 3. Then the Inner terms 4. Then the Last terms L

25 Method 2: FOIL (x + 2) (x + 5) 1. Multiply the First terms
Here are the steps: (x + 2) (x + 5) 1. Multiply the First terms

26 Method 2: FOIL (x + 2) (x + 5) 1. Multiply the First terms
Here are the steps: (x + 2) (x + 5) 1. Multiply the First terms

27 Method 2: FOIL Multiply x  x (x + 2) (x + 5)
Here are the steps: (x + 2) (x + 5) 1. Multiply the First terms Multiply x  x

28 Method 2: FOIL Multiply x  x x2 (x + 2) (x + 5)
Here are the steps: (x + 2) (x + 5) 1. Multiply the First terms Multiply x  x x2

29 Method 2: FOIL x2 (x + 2) (x + 5) 2. Then the Outer terms
1. Multiply the First terms 2. Then the Outer terms x2

30 Method 2: FOIL x2 (x + 2) (x + 5) 2. Then the Outer terms
1. Multiply the First terms 2. Then the Outer terms x2

31 Method 2: FOIL Multiply x  5 x2 (x + 2) (x + 5)
1. Multiply the First terms Multiply x  5 2. Then the Outer terms x2

32 Method 2: FOIL Multiply x  5 x2 + 5x (x + 2) (x + 5)
1. Multiply the First terms Multiply x  5 2. Then the Outer terms x2 + 5x

33 Method 2: FOIL x2 + 5x (x + 2) (x + 5) 3. Then the Inner terms
1. Multiply the First terms 2. Then the Outer terms 3. Then the Inner terms x2 + 5x

34 Method 2: FOIL x2 + 5x (x + 2) (x + 5) 3. Then the Inner terms
1. Multiply the First terms 2. Then the Outer terms 3. Then the Inner terms x2 + 5x

35 Method 2: FOIL Multiply 2  x x2 + 5x (x + 2) (x + 5)
1. Multiply the First terms Multiply 2  x 2. Then the Outer terms 3. Then the Inner terms x2 + 5x

36 Method 2: FOIL Multiply 2  x x2 + 5x + 2x (x + 2) (x + 5)
1. Multiply the First terms Multiply 2  x 2. Then the Outer terms 3. Then the Inner terms x2 + 5x + 2x

37 Method 2: FOIL x2 + 5x + 2x (x + 2) (x + 5) 4. Then the Last terms
1. Multiply the First terms 2. Then the Outer terms 3. Then the Inner terms x2 + 5x + 2x 4. Then the Last terms

38 Method 2: FOIL x2 + 5x + 2x (x + 2) (x + 5) 4. Then the Last terms
1. Multiply the First terms 2. Then the Outer terms 3. Then the Inner terms x2 + 5x + 2x 4. Then the Last terms

39 Method 2: FOIL Multiply 2  5 x2 + 5x + 2x (x + 2) (x + 5)
1. Multiply the First terms Multiply 2  5 2. Then the Outer terms 3. Then the Inner terms x2 + 5x + 2x 4. Then the Last terms

40 Method 2: FOIL Multiply 2  5 x2 + 5x + 2x + 10 (x + 2) (x + 5)
1. Multiply the First terms Multiply 2  5 2. Then the Outer terms 3. Then the Inner terms x2 + 5x + 2x + 10 4. Then the Last terms

41 Method 2: FOIL x2 + 5x + 2x + 10 (x + 2) (x + 5) Combine Like Terms
1. Multiply the First terms 2. Then the Outer terms 3. Then the Inner terms x2 + 5x + 2x + 10 4. Then the Last terms Combine Like Terms

42 Method 2: FOIL x2 + 5x + 2x + 10 (x + 2) (x + 5) Combine Like Terms
1. Multiply the First terms 2. Then the Outer terms 3. Then the Inner terms x2 + 5x + 2x + 10 4. Then the Last terms Combine Like Terms

43 Method 2: FOIL x2 + 5x + 2x + 10 (x + 2) (x + 5) Combine Like Terms
1. Multiply the First terms 2. Then the Outer terms 3. Then the Inner terms x2 + 5x + 2x + 10 4. Then the Last terms Combine Like Terms + 7x

44 Method 2: FOIL x2 + 5x + 2x + 10 (x + 2) (x + 5) + 7x
1. Multiply the First terms 2. Then the Outer terms 3. Then the Inner terms x2 + 5x + 2x + 10 4. Then the Last terms + 7x

45 Method 2: FOIL x2 + 5x + 2x + 10 (x + 2) (x + 5) x2 + 7x
1. Multiply the First terms 2. Then the Outer terms 3. Then the Inner terms x2 + 5x + 2x + 10 4. Then the Last terms x2 + 7x

46 Method 2: FOIL x2 + 5x + 2x + 10 (x + 2) (x + 5) x2 + 7x
1. Multiply the First terms 2. Then the Outer terms 3. Then the Inner terms x2 + 5x + 2x + 10 4. Then the Last terms x2 + 7x

47 Method 2: FOIL + 5x + 2x + 10 (x + 2) (x + 5) x2 + 7x + 10
1. Multiply the First terms 2. Then the Outer terms 3. Then the Inner terms + 5x + 2x + 10 4. Then the Last terms x2 + 7x + 10

48 Method 2: FOIL x2 + 5x + 2x + 10 (x + 2) (x + 5) x2 + 7x + 10
1. Multiply the First terms 2. Then the Outer terms 3. Then the Inner terms x2 + 5x + 2x + 10 4. Then the Last terms x2 + 7x + 10

49 Method 3: Algebra Tiles (x + 2) (x + 5)

50 Method 3: Algebra Tiles (x + 2) (x + 5)
1. Place the first factor on the top.

51 Method 3: Algebra Tiles (x + 2) (x + 5)
1 1 1. Place the first factor on the top.

52 Method 3: Algebra Tiles (x + 2) (x + 5)
1 1 1. Place the first factor on the top. 2. Place the second factor on the side.

53 Method 3: Algebra Tiles (x + 2) (x + 5)
1 1 1. Place the first factor on the top. x 2. Place the second factor on the side. x + 5 1 1 1 1 1

54 Method 3: Algebra Tiles (x + 2) (x + 5)
1 1 1. Place the first factor on the top. x 2. Place the second factor on the side. x 3. The product will fill in the rectangular area formed by the two sides. Maintain straight sides. + 5 1 1 1 1 1

55 Method 3: Algebra Tiles (x + 2) (x + 5)
1 1 1. Place the first factor on the top. x 2. Place the second factor on the side. x 3. The product will fill in the rectangular area formed by the two sides. Maintain straight sides. + 5 1 1 1 1 1

56 Method 3: Algebra Tiles (x + 2) (x + 5)
1 1 1. Place the first factor on the top. x2 x 2. Place the second factor on the side. x 3. The product will fill in the rectangular area formed by the two sides. Maintain straight sides. + 5 1 1 1 1 1

57 Method 3: Algebra Tiles (x + 2) (x + 5)
1 1 1. Place the first factor on the top. x2 x x x 2. Place the second factor on the side. x 3. The product will fill in the rectangular area formed by the two sides. Maintain straight sides. + 5 1 x 1 x 1 x 1 x 1 x

58 Method 3: Algebra Tiles (x + 2) (x + 5)
1 1 1. Place the first factor on the top. x2 x x x 2. Place the second factor on the side. x 3. The product will fill in the rectangular area formed by the two sides. Maintain straight sides. + 5 1 x 1 1 1 x 1 1 1 x 1 1 1 x 1 1 1 x 1 1

59 Method 3: Algebra Tiles (x + 2) (x + 5) 4. Read the product. x + 2 x 1
1. Place the first factor on the top. x2 x x x 2. Place the second factor on the side. x 3. The product will fill in the rectangular area formed by the two sides. Maintain straight sides. + 5 1 x 1 1 1 x 1 1 1 x 1 1 4. Read the product. 1 x 1 1 1 x 1 1

60 Method 3: Algebra Tiles (x + 2) (x + 5) 4. Read the product. x + 2 x 1
1. Place the first factor on the top. x2 x x x 2. Place the second factor on the side. x 3. The product will fill in the rectangular area formed by the two sides. Maintain straight sides. + 5 1 x 1 1 1 x 1 1 1 x 1 1 4. Read the product. 1 x 1 1 1 x 1 1

61 x2 Method 3: Algebra Tiles (x + 2) (x + 5) 4. Read the product. x + 2
1 1 1. Place the first factor on the top. x2 x x x 2. Place the second factor on the side. x 3. The product will fill in the rectangular area formed by the two sides. Maintain straight sides. + 5 1 x 1 1 1 x 1 1 1 x 1 1 4. Read the product. 1 x 1 1 x2 1 x 1 1

62 x2 + 7x Method 3: Algebra Tiles (x + 2) (x + 5) 4. Read the product.
1 1 1. Place the first factor on the top. x2 x x x 2. Place the second factor on the side. x 3. The product will fill in the rectangular area formed by the two sides. Maintain straight sides. + 5 1 x 1 1 1 x 1 1 1 x 1 1 4. Read the product. 1 x 1 1 x2 + 7x 1 x 1 1

63 x2 + 7x + 10 Method 3: Algebra Tiles (x + 2) (x + 5)
1. Place the first factor on the top. x2 x x x 2. Place the second factor on the side. x 3. The product will fill in the rectangular area formed by the two sides. Maintain straight sides. + 5 1 x 1 1 1 x 1 1 1 x 1 1 4. Read the product. 1 x 1 1 x2 + 7x + 10 1 x 1 1

64 x2 + 7x + 10 Method 3: Algebra Tiles (x + 2) (x + 5) x + 2 1 x x2 x x

65 Method 4: The Generic Rectangle
Instead of drawing the individual algebra tiles, you will just draw an simple rectangle to represent the area formed by the tiles. (x + 2) (x + 5)

66 Method 4: The Generic Rectangle
(x + 2) (x + 5) 1. Draw a rectangle and subdivide it.

67 Method 4: The Generic Rectangle
(x + 2) (x + 5) 1. Draw a rectangle and subdivide it.

68 Method 4: The Generic Rectangle
(x + 2) (x + 5) 1. Draw a rectangle and subdivide it. 2. Place the 1st factor on the top and the 2nd factor on the side.

69 Method 4: The Generic Rectangle
(x + 2) (x + 5) x + 2 1. Draw a rectangle and subdivide it. x 2. Place the 1st factor on the top and the 2nd factor on the side. + 5

70 Method 4: The Generic Rectangle
(x + 2) (x + 5) x + 2 1. Draw a rectangle and subdivide it. x 2. Place the 1st factor on the top and the 2nd factor on the side. + 5 3. Multiply terms to fill in the area.

71 Method 4: The Generic Rectangle
(x + 2) (x + 5) x + 2 1. Draw a rectangle and subdivide it. x 2. Place the 1st factor on the top and the 2nd factor on the side. + 5 3. Multiply terms to fill in the area. x  x =

72 Method 4: The Generic Rectangle
(x + 2) (x + 5) x + 2 1. Draw a rectangle and subdivide it. x2 x 2. Place the 1st factor on the top and the 2nd factor on the side. + 5 3. Multiply terms to fill in the area.

73 Method 4: The Generic Rectangle
(x + 2) (x + 5) x + 2 1. Draw a rectangle and subdivide it. x2 x 2. Place the 1st factor on the top and the 2nd factor on the side. + 5 3. Multiply terms to fill in the area.

74 Method 4: The Generic Rectangle
(x + 2) (x + 5) x + 2 1. Draw a rectangle and subdivide it. x2 x 2. Place the 1st factor on the top and the 2nd factor on the side. + 5 3. Multiply terms to fill in the area. x  2 =

75 Method 4: The Generic Rectangle
(x + 2) (x + 5) x + 2 1. Draw a rectangle and subdivide it. x2 2x x 2. Place the 1st factor on the top and the 2nd factor on the side. + 5 3. Multiply terms to fill in the area.

76 Method 4: The Generic Rectangle
(x + 2) (x + 5) x + 2 1. Draw a rectangle and subdivide it. x2 2x x 2. Place the 1st factor on the top and the 2nd factor on the side. + 5 3. Multiply terms to fill in the area.

77 Method 4: The Generic Rectangle
(x + 2) (x + 5) x + 2 1. Draw a rectangle and subdivide it. x2 2x x 2. Place the 1st factor on the top and the 2nd factor on the side. + 5 3. Multiply terms to fill in the area. 5  x =

78 Method 4: The Generic Rectangle
(x + 2) (x + 5) x + 2 1. Draw a rectangle and subdivide it. x2 2x x 2. Place the 1st factor on the top and the 2nd factor on the side. 5x + 5 3. Multiply terms to fill in the area.

79 Method 4: The Generic Rectangle
(x + 2) (x + 5) x + 2 1. Draw a rectangle and subdivide it. x2 2x x 2. Place the 1st factor on the top and the 2nd factor on the side. 5x + 5 3. Multiply terms to fill in the area.

80 Method 4: The Generic Rectangle
(x + 2) (x + 5) x + 2 1. Draw a rectangle and subdivide it. x2 2x x 2. Place the 1st factor on the top and the 2nd factor on the side. 5x + 5 3. Multiply terms to fill in the area. 5  2 =

81 Method 4: The Generic Rectangle
(x + 2) (x + 5) x + 2 1. Draw a rectangle and subdivide it. x2 2x x 2. Place the 1st factor on the top and the 2nd factor on the side. 5x + 5 10 3. Multiply terms to fill in the area.

82 Method 4: The Generic Rectangle
(x + 2) (x + 5) x + 2 1. Draw a rectangle and subdivide it. x2 x2 2x 2x x 2. Place the 1st factor on the top and the 2nd factor on the side. 5x 5x + 5 10 10 3. Multiply terms to fill in the area. 4. Read the product and combine like terms.

83 Method 4: The Generic Rectangle
(x + 2) (x + 5) x + 2 1. Draw a rectangle and subdivide it. x2 x2 2x 2x x 2. Place the 1st factor on the top and the 2nd factor on the side. 5x 5x + 5 10 10 3. Multiply terms to fill in the area. x2 4. Read the product and combine like terms.

84 Method 4: The Generic Rectangle
(x + 2) (x + 5) x + 2 1. Draw a rectangle and subdivide it. x2 x2 2x 2x x 2. Place the 1st factor on the top and the 2nd factor on the side. 5x 5x + 5 10 10 3. Multiply terms to fill in the area. x2 + 7x 4. Read the product and combine like terms.

85 Method 4: The Generic Rectangle
(x + 2) (x + 5) x + 2 1. Draw a rectangle and subdivide it. x2 x2 2x 2x x 2. Place the 1st factor on the top and the 2nd factor on the side. 5x 5x + 5 10 10 3. Multiply terms to fill in the area. x2 + 7x + 10 4. Read the product and combine like terms.

86 Method 4: The Generic Rectangle
(x + 2) (x + 5) x + 2 1. Draw a rectangle and subdivide it. x2 x2 2x 2x x 2. Place the 1st factor on the top and the 2nd factor on the side. 5x 5x + 5 10 10 3. Multiply terms to fill in the area. x2 + 7x + 10 4. Read the product and combine like terms.

87 What were the advantages of each method?
The Vertical Method FOIL Algebra Tiles The Generic Rectangle

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