Download 4(x + 2)

The Distributive Property
and Factoring
Day 2
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?
4
7
3
4
7+ 3
You could multiply 4 by 7,
then 4 by 3 and add them
You could add 7 + 3
and then multiply by 4
OR
4(7+3)=
4(10)=
40
4(7) + 4(3) =
28 + 12 =
40
2
Either way, the area is 40 feet :
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.
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+ 3
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
Finding the area of the rectangles demonstrates the
distributive property. Use the distributive property when
expressions are
written like so: a(b + c)
4(x + 2)
4(x) + 4(2)
4x + 8
The 4 is distributed to each term of the sum (x + 2)
Definition: The Distributive Property
Of Addition lets you multiply
a sum by multiplying each addend
separately and then add the
products.
There are two distributive properties:
a) Distributive Property of Addition
Ex.: 4(7 + 3) = 4(7) + 4(3) = 28 + 12 = 40
or 4(10) = 40
b) Distributive Property of Subtraction
Ex.: 5(3 - 6) = 5(3) - 5(6) = 15 - 30 = -15
or 5 (-3) = -15
Write an expression equivalent to:
5(y + 4)
5(y) + 5(4)
5y + 20
Remember to distribute the 5 to the y and the 4
6(x + 2)
3(x + 4)
4(x - 5)
7(x - 1)
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.
Be careful with
EXAMPLE:
your signs!
-2(x + 3) = -2(x) + -2(3) = -2x + -6 or -2x - 6
3(4x - 6) = 3(4x) - 3(6) = 12x - 18
-2 (x - 3) = -2(x) - (-2)(3) = -2x + 6
TRY THESE:
3(4x + 2) =
-1(6m + 4) =
-3(2x - 5) =
Keep in mind that when there is a negative sign on the
outside of the parenthesis it really is a -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?
Remove to see answer.
The numbers are turned to their opposites.
Try these:
-(9x + 3) =
-(-5x + 1) =
-(2x - 4) =
-(-x - 6) =
20
4(2 + 5) = 4(2) + 5
A
B
True
False
21
8(x + 9) = 8(x) + 8(9)
A
B
True
False
22
-4(x + 6) = -4 + 4(6)
A
B
True
False
23
3(x - 4) = 3(x) - 3(4)
A
B
True
False
24 Use the distributive property to rewrite the
expression without parentheses
3(x + 4)
A
B
C
D
3x + 4
3x + 12
x + 12
7x
25 Use the distributive property to rewrite the
expression without parentheses
5(x + 7)
A
B
C
D
x + 35
5x + 7
5x + 35
40x
26 Use the distributive property to rewrite the
expression without parentheses
(x + 5)2
A
B
C
D
2x + 5
2x + 10
x + 10
12x
27 Use the distributive property to rewrite the
expression without parentheses
3(x - 4)
A
B
C
D
3x - 4
x - 12
3x - 12
9x
28 Use the distributive property to rewrite the
expression without parentheses
2(w - 6)
A
B
C
D
2w - 6
w - 12
2w - 12
10w
29 Use the distributive property to rewrite the
expression without parentheses
-4(x - 9)
A
B
C
D
-4x - 36
x - 36
4x - 36
-4x + 36
30 Use the distributive property to rewrite the
expression without parentheses
5.2(x - 9.3)
A
B
C
D
-5.2x - 48.36
5.2x - 48.36
-5.2x + 48.36
-48.36x
31 Use the distributive property to rewrite the
expression without parentheses
A
B
C
D
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 + 35 Both 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 = 2(x + 2y)
9b + 3 = 3(3b + 1)
-5j - 10k + 25m = -5(j + 2k - 5m) *Careful of your signs
4a + 6a + 8ab = 2a(2 + 3 + 4b)
Try these:
Factor the following expressions:
1.) 6b + 9c =
2.) -2h - 10j =
3.) 4a + 20ab + 12abc =
32 Factor the following: 4p + 24q
A
B
C
D
4 (p + 24q)
2 (2p + 12q)
4(p + 6q)
2 (2p + 24q)
33 Factor the following: 5g + 15h
A
B
3(g + 5h)
5(g + 3h)
C
D
5(g + 15h)
5g (1 + 3h)
34 Factor the following: 3r + 9rt + 15rx
A
B
C
D
3(r+ 3rt + 5rx)
3r(1 + 3t + 5x)
3r (3t + 5x)
3 (r + 9rt + 15rx)
35 Factor the following: 2v+7v+14v
A
B
C
D
7(2v + v + 2v)
7v(2 + 1 + 2)
7v (1 + 2)
v(2 + 7 + 14)
36 Factor the following: -6a - 15ab - 18abc
A
B
C
D
-3a(2 + 5b + 6bc)
3a(2+ 5b + 6bc)
-3(2a - 5b - 6bc)
-3a (2 -5b - 6bc)
-
What divides into the expression: -5n - 20mn - 10np
-
If a regular pentagon has a perimeter of 10x + 25,
what does each side equal?