Ch 2.6 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) 5) (3 - k)5 6) x(a + m) 7) -4(3 - r) 8) 2(x - 8) 9) -1(2m - 3) 10) (6 - 2y)3 3x a m a k ax + mx r 2x m y

(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 4y b m a k ac + ad 3 + r 4x m y -3y

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

Combine like terms. S IMPLIFYING BY C OMBINING L IKE T ERMS 4x – x 2 = (8 + 3)x Use the distributive property. = 11x Add coefficients. 8x + 3x Group like terms. Rewrite as addition expression. Distribute the –2. Multiply. Combine like terms and simplify 4x 2 – x = 3x – 2(4 + x) = 3 + (–2)(4 + x) = 3 + [(–2)(4) + (–2)(x)] = 3 + (–8) + (–2x) = –5 + (–2x) = –5 – 2x = Designate one sign in front of 2x

Subtracting a Quantity 1) -(x + 6) 2) -(2x - 8) 3) 10- (4m + 3) 4) 2(x - 5) - (x - 3) 5) -(3a + 1) 6) -(-3x + 2x -7) 7) (3y - 8) 8) 4(3k - 5) - (2k + 9) -x x m m + 7 2x x + 3 x a x - 2x y y k k k

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) =