Lesson 8-3 Warm-Up

“Multiplying Binomials” (8-3) What is a binomial? How do you multiply two binomials together? Binomial: a polynomial that is the sum or difference of two terms that are not like terms Examples: 2x + 1 9x4 + 11x -5x3 - 7 Method 1: Use an Area Model (write the product of each row and column in the box where they intersect, or cross) Example: Simplify (2x + 3)(x + 4) Create an area model in which each term has it own row or column. Write the product of intersecting terms in the box where they intersect 2x2 + 8x + 3x + 12 Add the answers to each box 2x2 + 11x + 12 Combine like terms. 2x + 3 2x • x = 2x2 3 • x = 3x 2x • 4 = 8x 3 • 4 = 12 x + 4

“Multiplying Binomials” (8-3) Method 2: Use the Distributive Property (horizontal method) Example: Simplify (2x + 3)(x + 4) (2x + 3)(x + 4) (2x • x) + (2x • 4) + 3x + 12 2x2 + 8x + 3x + 12 2x2 + 11x + 12 Combine like terms. Tip: An easy way to remember what to multiply when using the Distributive Property to multiply binomials is to use FOIL, which is a mnemonic device (memory tool) like PEMDAS that stands for “First, Outer, Inner, Last”. Example: Simplify (3x - 5)(2x + 7) First + Outer + Inner + Last (3x - 5)(2x + 7) + (3x - 5)(2x + 7) + (3x - 5)(2x + 7) + (3x - 5)(2x + 7) 6x2 + 21x + -10x + -35 6x2 + 11x - 35 Multiply the first term of the first binomail by each term of the second binomial. Then, multiply the second term of the first binomial by each term of the second binomial. Add these product together (don’t forget to combine like terms)

“Multiplying Binomials” (8-3) Method 3: Use the Distributive Property (vertical method): Multiply the two binomials together treating each term as a place value Example: Simplify (2x + 3)(x + 4) 2x + 3 x x + 4 8x + 12 2x2 + 3x 2x2 + 11x + 12

“Multiplying Binomials” (8-3) Method 4: Use an Area Model: Arrange the Algebra Tiles that model the binomial into the length and width of a rectangle. Then, find the area of the rectangle using Algebra Tiles (how many of each type it would take to completely fill the rectangle made by the length and width). Example: Simplify (2x + 3)(x + 4) (2x + 3)(x + 4) = 2x2 + 11x + 12 x x 1 1 1 Algebra Tiles Key x 1 1 x 1 x2 x2 x x x x x2 x x 1 x x 1 1 1 1 x x 1 1 1 1 x x 1 1 1 1 x x 1 1 1 x 1

2y -3 y 2y2 -3y 2 4y -6 Simplify (2y – 3)(y + 2). Multiplying Binomials LESSON 8-3 Additional Examples Simplify (2y – 3)(y + 2). (2y – 3)(y + 2) = (2y – 3)(y) Distribute 2y – 3. + (2y – 3)(2) = 2y2 – 3y Now distribute y and 2. + 4y – 6 = 2y2 + y – 6 Simplify. 1. Set up a table 2y -3 2. Multiply terms in where rows and columns cross. y 2y2 -3y 3. Combine like terms. 2 4y -6 The product is 2y2 + y - 6.

Simplify (4x + 2)(3x – 6). First = (4x)(3x) (4x + 2)(3x – 6) Outer Multiplying Binomials LESSON 8-3 Additional Examples Simplify (4x + 2)(3x – 6). First = (4x)(3x) (4x + 2)(3x – 6) Outer (4x)(–6) + Last (2)(–6) + Inner (2)(3x) + 24x 6x 12 – + = 12x2 = 12x2 18x – 12 The product is 12x2 – 18x – 12.

Find an expression for the area of the shaded region. Simplify. Multiplying Binomials LESSON 8-3 Additional Examples Find an expression for the area of the shaded region. Simplify. area of outer rectangle (includes white area) = (3x + 2)(2x – 1) area of inner rectangle (white area) = x(x + 3) area of shaded region = area of outer rectangle – area of inner rectangle = (3x + 2)(2x – 1) x(x + 3) Given areas. x2 + 3x = 6x2 Use FOIL to simplify (3x + 2) (2x – 1) and the Distributive Property to simplify x(x + 3). – 3x + 4x – 2 = 6x2 – 3x + 4x – 2 - (x2 + 3x) Shaded Area = Outer Area – Inner (White) Area = 6x2 – 3x + 4x – 2 Group like terms. – x2 – 3x = 5x2 – 2x – 2 Simplify.

Simplify the product (3x2 – 2x + 3)(2x + 7). Multiplying Binomials LESSON 8-3 Additional Examples Simplify the product (3x2 – 2x + 3)(2x + 7). Method 1:  Multiply using the vertical method. 3x2  –   2x  +  3 x 2x  +  7 21x2  –  14x  +  21   Multiply by 7. 6x3  –  4x2 +    6x Multiply by 2x. 6x3  + 17x2  –   8x  +  21 Add like terms.

3x2 -2x 3 2x 6x3 -4x2 6x 7 21x2 -14x 21 (continued) Multiplying Binomials LESSON 8-3 Additional Examples (continued) Method 2:  Multiply using the horizontal method. (2x + 7)(3x2 – 2x + 3) = (2x)(3x2) – (2x)(2x) + (2x)(3) + (7)(3x2) – (7)(2x) + (7)(3) = 6x3 – 4x2 + 6x + 21x2 – 14x + 21 = 6x3 + -4x2 + 21x2 + 6x + (-14x) + 21 Combine like terms. 3x2 -2x 3 2x 6x3 -4x2 6x The product is 6x3 + 17x2 – 8x + 21. 7 21x2 -14x 21

Simplify each product using any method. Multiplying Binomials LESSON 8-3 Lesson Quiz Simplify each product using any method. 1. (x + 3)(x – 6) 2. (2b – 4)(3b – 5) 3. (3x – 4)(3x2 + x + 2) 4. Find the area of the shaded region. x2 – 3x – 18 6b2 – 22b + 20 9x3 – 9x2 + 2x – 8 2x2 + 3x – 1