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The Distributive Property Algebra; Section 2.6 Goal 1: Using the Distributive Property ¢ Rules l The Distributive Property Example 1: Using an Area Model ¢ Find the area of a rectangle whose width is 3 and whose length is x + 2. ¢ Solution l Draw a picture: 3 3 3x 6 x+2 x 2 Example 1: Using an Area Model ¢ Solution l Draw a picture: 3 3 3x 6 x+2 x 2 l The picture would lead us to believe that the area is 3x + 6. The Distributive Property ¢ a(b + c) = ab + ac. ¢ a(b – c) = ab – ac. ¢ The factor on the outside gets distributed to both of the terms on the inside. Example 2: Using the Distributive Property ¢ Find the following products: l 2(x + 5) l (x – 4)x l (1 + 2x)8 l y(1 – y) Example 2: Using the Distributive Property ¢ Solution: l 2(x + 5) = 2x + 2(5) = 2x + 10 l (x – 4)x = x2 – 4x l (1 + 2x)8 = 8(1) + 8(2x) = 8 + 16x l y(1 – y) = 1y – y2 = y – y2 Example 3: Using the Distributive Property ¢ Find the following products: l -3(x + 4) l (y + 5)(-4) l -(6 – 3x) l (x – 1)(-9x) Example 3: Using the Distributive Property ¢ Solution: l -3(x + 4) = -3x + (-3)(4) = -3x – 12 l (y + 5)(-4) = -4y + (-4)(5) = -4y – 20 l -(6 – 3x) = -6 – -(3x) = -6 + 3x l (x – 1)(-9x) = -9x2 – 1(-9x) = -9x2 + 9x Example 4: Mental Math Calculations ¢ You are shopping for CDs. You want to buy 6 CDs for $11.95 each. Use the Distributive Property to calculate the total cost. ¢ Solution l Think of $11.95 as $12.00 – $.05. l 6(12 – .05) = 6(12) – 6(.05) l 6(12) – 6(.05) = 72.00 – .30 = $71.70 Extra Example: Mental Math Calculations ¢ Find the following products: l 29(30) l 41(11) ¢ Solution l 29(30) = (30 – 1)(30) = 900 – 30 = 870 l 41(11) = 41(10 + 1) = 410 + 41 = 451 In-Class Assignment ¢ Do #3-12 on page 103. Goal 2: Simplifying by Combining Like Terms ¢ Vocabulary l Coefficient l Like terms l Constant terms l Simplified Example 5: Simplifying by Combining Like Terms ¢ Simplify the following: l 8x + 3x l 4x2 + 2 – x2 l 3 – 2(4 + x) ¢ Solution l 8x + 3x = (8 + 3)x = 11x l 4x2 + 2 – x2 = (4 – 1)x2 + 2 = 3x2 + 2 l 3 – 2(4 + x) = 3 – 2(4) – 2(x) = 3 – 8 – 2x = -5 – 2x Example 6: Using the Distributive Property to Simplify a Function ¢ It takes you 45 minutes to get to school. You spend t minutes walking to the bus stop, and then the rest of the time riding the bus. You walk .06 miles/minute and the bus travels .5 miles/minutes. The total distance you travel is given by the function D = .06t + .5(45 – t). Simplify this function. Example 6: Using the Distributive Property to Simplify a Function ¢ Solution: l D = .06t + .5(45 – t) = .06t + .5(45) + .5(-t) l .06t + .5(45) + .5(-t) = .06t + 22.5 – .5t l .06t + 22.5 – .5t = -.44t + 22.5 In-Class Assignment ¢ Do #1,2, and #13-18 on page 103. Assignment ¢ Do #19-24 all, #26-68 evens, and #70-78 all on pages 103-104.