Unit 2, Lesson 2: The Distributive Property and Factoring

An Area Model Imagine that you have two rooms next to each other. Both are 4 feet long. One is 7 feet wide and the other is 3 feet wide How could you express the area of those two rooms together?

Either way, the area is 40 feet 2 : You could add and then multiply by 4 4(7+3)= 4(10)= 40 OR You could multiply 4 by 7, then 4 by 3 and add them 4(7) + 4(3) = = 40

An Area Model Imagine that you have two rooms next to each other. Both are 4 yards long. One is 3 yards wide and you don't know how wide the other is. 4 x 3 How could you express the area of those two rooms together?

4 x + 3 You cannot add x and 3 because they aren't like terms, so you can only do it by multiplying 4 by x and 4 by 3 and adding 4(x) + 4(3)= 4x + 12 The area of the two rooms is 4x + 12 (Note: 4x cannot be combined with 12)

The Distributive Property Distributive Property: a(b + c) = ab + ac 4(x + 2) 4(x) + 4(2) 4x + 8 The multiplication of 4 is distributed to each term of the sum (x + 2).

Write an expression equivalent to: 5(y + 4)= 5(y) + 5(4)= 5y + 20= 6(x + 2)

The Distributive Property is often used to eliminate the parentheses in expressions like 4(x + 2). This makes it possible to combine like terms in more complicated expressions. EXAMPLES: -2(x + 3) = 3(4x - 6) = -2 (x - 3) = Be careful with your signs!

TRY THESE: 1) 3(4x + 2) = 2) -1(6m + 4) = 3) -3(2x - 5) =

Keep in mind that when there is a negative sign on the outside of the parenthesis it really is a -1 (an implied -1). For example: -(2x + 7) = -1(2x + 7) = -1(2x) + -1(7) = -2x - 7 What do you notice about the original problem and its answer? The numbers are turned to their opposites. Remove to see answer. Try these: -(9x + 3) = -(-5x + 1) = -(2x - 4) = -(-x - 6) =

We can also use the Distributive Property in reverse. This is called Factoring. When we factor an expression, we find all numbers or variables that divide into all of the parts of an expression. Example: 7x + 35Both the 7x and 35 are divisible by 7 7(x + 5)By removing the 7 we have factored the problem We can check our work by using the distributive property to see that the two expressions are equal.

We can factor with numbers, variables, or both. 2x + 4y = -5j - 10k + 25m = 4a + 6a + 8ab =

Try these: Factor the following expressions: 1.) 6b + 9c = 2.) -2h - 10j = 3.) 4a + 20ab + 12abc =

If a regular pentagon has a perimeter of 10x + 25, what does each side equal?

21 8(x + 9) = 8(x) + 8(9) A True B False

22 -4(x + 6) = (6) A True B False

24Use the distributive property to rewrite the expression without parentheses 3(x + 4) A3x + 4 B3x + 12 Cx + 12 D7x

26Use the distributive property to rewrite the expression without parentheses (x + 5)2 A2x + 5 B2x + 10 Cx + 10 D12x

27Use the distributive property to rewrite the expression without parentheses 3(x - 4) A3x - 4 Bx - 12 C3x - 12 D9x

29Use the distributive property to rewrite the expression without parentheses -4(x - 9) A-4x - 36 Bx - 36 C4x - 36 D-4x + 36

30Use the distributive property to rewrite the expression without parentheses 5.2(x - 9.3) A-5.2x B5.2x C-5.2x D-48.36x

31Use the distributive property to rewrite the expression without parentheses A B C D

32Factor the following: 4p + 24q A4 (p + 24q) B2 (2p + 12q) C4(p + 6q) D2 (2p + 24q)

33Factor the following: 5g + 15h A3(g + 5h) B5(g + 3h) C5(g + 15h) D5g (1 + 3h)

34Factor the following: 3r + 9rt + 15rx A3(r+ 3rt + 5rx) B3r(1 + 3t + 5x) C3r (3t + 5x) D3 (r + 9rt + 15rx)

36Factor the following: -6a - 15ab - 18abc A-3a(2 + 5b + 6bc) B3a(2+ 5b + 6bc) C-3(2a - 5b - 6bc) D-3a (2 -5b - 6bc)