# 1-5 The distributive Property

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1-5 The distributive Property
You can use algebra tiles to model algebraic expressions. 1 1-tile 1 x-tile 1 x This 1-by-1 square tile has an area of 1 square unit. This 1-by-x square tile has an area of x square units. Model the Distributive Property using Algebra Tiles 3 3 x 3 2 x + 2 + x + 2 Area = 3(x + 2) Area = 3(x ) + 3(2)

a(b + c) = ab + ac 2(x + 5) = 2(x) + 2(5) = 2x + 10 (b + c)a = ba + ca
USE THE DISTRIBUTIVE PROPERTY THE DISTRIBUTIVE PROPERTY The product of a and (b + c): a(b + c) = ab + ac 2(x + 5) = 2(x) + 2(5) = 2x + 10 (b + c)a = ba + ca (x + 5)2 = (x)2 + (5)2 = 2x + 10 y(1 – y) = y(1) – y(y) = y – y 2 = = (1 + 5x)2 (1)2 + (5x)2 2 + 10x

(–3)(1 + x) = (–3)(1) + (–3)(x) = –3 – 3x (y – 5)(–2)
USING THE DISTRIBUTIVE PROPERTY Remember that a factor must multiply each term of an expression. (–3)(1 + x) = (–3)(1) + (–3)(x) Distribute the –3. Simplify. = –3 – 3x (y – 5)(–2) = (y)(–2) + (–5)(–2) Distribute the –2. Simplify. = –2y + 10 –(7 – 3x) = (–1)(7) + (–1)(–3x) –a = –1 • a Simplify. = –7 + 3x Forgetting to distribute the negative sign when multiplying by a negative factor is a common error.

You are shopping for CDs. You want to buy six CDs for \$11.95 each.
MENTAL MATH CALCULATIONS SOLUTION You are shopping for CDs. You want to buy six CDs for \$11.95 each. The mental math is easier if you think of \$11.95 as \$12.00 – \$.05. 6(11.95) = 6(12 – 0.05) Write as a difference. = 6(12) – 6(0.05) Use the distributive property to calculate the total cost mentally. Use the distributive property. = 72 – 0.30 Find the products mentally. = 71.70 Find the difference mentally. The total cost of 6 CDs at \$11.95 each is \$71.70.

x is the y is the x2 y3 y2 – x2 + 3y3 – 5 + 3 – 3x2 + 4y3 + y + – 3 y2
SIMPLIFYING BY COMBINING LIKE TERMS Each of these terms is the product of a number and a variable. Each of these terms is the product of a number and a variable. terms + 3 y2 x variable. + 3 y2 x number + 3 y2 x + 3 y2 x x is the variable. y is the –1 is the coefficient of x. 3 is the coefficient of y2. variable power. Like terms Like terms have the same variable raised to the same power. y2 – x2 + 3y3 – – 3x2 + 4y3 + y The constant terms –5 and 3 are also like terms. x2 y3

8x + 3x = (8 + 3)x = 11x 4x2 + 2 – x2 = 4x2 – x2 + 2 = 3x2 + 2
SIMPLIFYING BY COMBINING LIKE TERMS 8x + 3x = (8 + 3)x Use the distributive property. = 11x Add coefficients. 4x2 + 2 – x2 = 4x2 – x2 + 2 Group like terms. = 3x2 + 2 Combine like terms. 3 – 2(4 + x) = 3 + (–2)(4 + x) Rewrite as addition expression. = 3 + [(–2)(4) + (–2)(x)] Distribute the –2. = 3 + (–8) + (–2x) Multiply. = –5 + (–2x) Combine like terms and simplify. = –5 – 2x