Ch 1.3 Distributive Property Objective: To use the distributive property to simplify variable expressions. Property Distributive Property The distributive property is used when multiplying an expression with a group of expressions that are added (or subtracted). For example: a(b + c) = a(b) + a(c) a(b - c) = a(b) - a(c) (b + c)a = (b)a + (c)a (b - c)a = (b)a - (c)a T HE D ISTRIBUTIVE P ROPERTY a(b + c) = ab + ac (b + c)a = ba + ca 2( x + 5 ) 2(x) + 2(5)2x + 10 (x + 5) 2 (x)2 + (5)2 2x + 10 ( 1 + 5x )2 (1)2 + (5x) x y(1 y) y(1) y(y) y y 2 = = == = = == The product of a and (b + c): U SE THE D ISTRIBUTIVE P ROPERTY Comparison Order of OperationsDistributive Property 6(3 + 5) 6(8) 48 6(3) + 6(5) Why distribute when order of operations is faster ? Use Distributive Property when there is a variable Use Order of Operation to check your answer Use the distributive property to simplify. 1) 3(x + 7) 2) 2(a - 4) 3) -7(8 - m) 4) -3(4 + a) 3x 2a Examples m - 3a Use the distributive property to simplify. 5) (3 - k)5 7) -4(3 - r) 6) - (2m - 3) 8) (6 - 2y) m 18 Examples - 5k 1 + 3m - 6y (y 5)(2) = (y)(2) + (5)(2) = 2y + 10 (7 3x) = (1)(7) + (1)(3x) = 7 + 3x = 3 3x (3)(1 + x) = (3)(1) + (3)(x) U SE THE D ISTRIBUTIVE P ROPERTY Remember that a factor must multiply EACH term of an expression. Forgetting to distribute the negative sign when multiplying by a negative factor is a common error. Use the distributive property to simplify. 1) 4(y - 7) 2) 3(b + 4) 3) -5(9 - m) 4) 5(4 - a) 5) (7 - k)6 6) a(c + d) 7) - (-3 - r) 8) 4(x - 8) 9) - (2m + 3) 10) (6 - 2y) -3y Classwork Find the difference mentally. Find the products mentally. The mental math is easier if you think of $11.95 as $12.00 $.05. Write as a difference. You are shopping for CDs. You want to buy six CDs for $11.95 each. Use the distributive property to calculate the total cost mentally. 6(11.95) = 6(12 0.05) Use the distributive property. = 6(12) 6(0.05) = 72 0.30 = The total cost of 6 CDs at $11.95 each is $ M ENTAL M ATH C ALCULATIONS Geometric Model for Area Two ways to find the total area. Width by total length (Order of Operations) Sum of smaller rectangles (Distributive Property) 4(3 + 7)4(3) + 4(7) 4(3)4(7) = 4 (10) = = 40 Geometric Model for Distributive Property 4 x 9 Two ways to find the total area. Width by total lengthSum of smaller rectangles 9(4 + x)9(4) + 9(x) =