Four Methods to Multiply Binomials (x + 2)(x + 5) Becky Afghani Long Beach Unified School District rafghani@lbusd.k12.ca.us 2006

What are the 4 methods?

What are the 4 methods? The Vertical Method

What are the 4 methods? The Vertical Method FOIL

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

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

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

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

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

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.

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

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

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.

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.

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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.

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

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

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

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

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

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

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

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

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

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

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

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

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)

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

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

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.

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

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.

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 =

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.

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.

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 =

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.

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.

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 =

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.

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.

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 =

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.

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.

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.

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.

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.

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.

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