Section 1.7 Properties of Real Numbers 73 ������������ Subtraction and division are not commutative. We cannot change the order of the numbers being subtracted or divided and obtain the same result. For instance, 5 - 3 = 2 but 3 - 5 = 3 + (-5) = -2. Also, 6 3 = 2 but 3 6 = 1 2 . The Distributive Property The distributive property is a property that is used extensively in algebra. It provides a way for us to multiply a group of numbers by a number—that is, it enables us to rewrite a product as a sum or a sum as a product. To illustrate, we will simplify the two expressions below using the order of operations. 3(4 + 5) = 3(9) 3(4) + 3(5) = 12 + 15 = 27 = 27 These two expressions are equivalent—that is, 3(4 + 5) = 3(4) + 3(5) The preceding example illustrates that when a number is multiplied by a sum, it is equivalent to the sum of the products. Also, consider a difference multiplied by a number. 3(4 - 5) = 3(-1) 3(4) - 3(5) = 12 - 15 = -3 = -3 These two expressions are equivalent, so 3(4 - 5) = 3(4) - 3(5) The preceding example illustrates that when a number is multiplied by a difference, it is equivalent to the difference of the products. These properties make up the distributive property over addition and over subtraction. �������������������� Distributive Property over Addition For all real numbers a, b, and c, a(b + c) = ab + ac �������������������� Distributive Property over Subtraction a(b - c) = ab - ac When asked to apply the distributive property, our goal is to multiply the factor outside of the parentheses by each of the terms inside the parentheses. Because multiplication is commutative, we can also distribute from the left and write the distributive property as follows. �������������������� Alternate Form of the Distributive Property (b + c)a = ba + ca An illustration of this alternate form is (2 + 6) 4 = 2(4) + 6(4) = 8 + 24 = 32. Objective 3 ▶ Apply the distributive property.
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