Chapter 2 – Properties of Real Numbers 2.6 – The Distributive Property

2.6 – The Distributive Property Today we will learn how to: Use the distributive property Simplify expressions by combining like terms

2.6 – The Distributive Property Recall: Multiplying can be modeled as repeated addition. 5(2) = 2 + 2 + 2 + 2 +2 5(2 + 4) = 2 +4 + 2 +4 + 2 + 4 + 2 + 4 + 2 + 4 5(2) + 5(4) This is an example of the DISTRIBUTIVE PROPERTY

2.6 – The Distributive Property The product of a and (b + c): a(b + c) = ab + ac Example: 5(x + 2) = 5x + 5(2) = 5x + 10 (b +c )a = ba + ca Example: (x + 4)8 = 8x + 8(4) = 8x + 32

2.6 – The Distributive Property The product of a and (b – c): a(b – c) = ab – ac Example: 4(x – 7) = 4x – 4(7) = 4x – 28 (b – c )a = ba – ca Example: (x – 5)9 = 9x – (5)9 = 9x – 45

2.6 – The Distributive Property Example 1: Simplify 3(x + 1) (x – 2)7 (2 + 5x)x d(2 – d)

2.6 – The Distributive Property Example 2: Simplify -6(x + 3) (t + 2)(-3) -1(1 – x) (2x – 3)(-4x)

2.6 – The Distributive Property Example 3 Find the area of a rectangle whose width is 4 and whose length is x + 2. x + 2 4

2.6 – The Distributive Property In a term that is the product of a number and a variable, the number is the COEFFICIENT of the variable. Example: In the expression –x + 3y2, -1 and 3 are the coefficients

2.6 – The Distributive Property LIKE TERMS are terms in an expression that have the same variable raised to the same power. Example: In the expression –x2 + 5x + (-4) + (-3x) + 2, 5x and -3x are like terms; -4 and 2 are like terms -4 and 2 are also CONSTANT TERMS

2.6 – The Distributive Property The distributive property allows you to combine like terms that have variables by adding coefficients. An expression is SIMPLIFIED if it has: No grouping symbols All of the like terms have been combined

2.6 – The Distributive Property Example 5 Simplify the expression 4x – x 5x2 – 7 + 3x2 1 – 4(1 – 2x)

2.6 – The Distributive Property Example 6 If takes you 30 minutes to get to school. You spend t minutes walking to the bus stop, and the rest of the time riding the bus. You walk 0.04 miles/minute and the bus travel 0.8 miles/minute. The total distance you travel is given by the function D = 0.04t + 0.8(30 – t). Simplify this function.

2.6 – The Distributive Property HOMEWORK Page 103 #40 – 48, 56 – 71, 79 – 80