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
USE THE DISTRIBUTIVE PROPERTY
THE DISTRIBUTIVE PROPERTY
The product of a and (b + c):
= 2(x) + 2(5) = 2x + 10
a(b + c) = ab + ac
2(x + 5)
(b + c)a = ba + ca
(x + 5)2 = (x)2 + (5)2 = 2x + 10
y(1 – y)
= y(1) – y(y)
2
y
–
y
=
(1 + 5x)2
= (1)2 + (5x)2
= 2 + 10x
Comparison
Order of Operations
6(3 + 5)
Distributive Property
6(3 + 5)
6(8)
6(3) + 6(5)
48
18 + 30
48
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)
6) x(a + m)
3x + 21
ax + mx
2) 2(a - 4)
7) -4(3 - r)
2a - 8
-12 + 4r
3) -7(8 - m)
8) 2(x - 8)
-56 + 7m
2x - 16
4) 3(4 - a)
9) -1(2m - 3)
12 - 3a
-2m + 3
5) (3 - k)5
10) (6 - 2y)3
15 - 5k
18 - 6y
USE THE DISTRIBUTIVE PROPERTY
Remember that a factor must multiply EACH term of an expression.
(–3)(1 + x)
= (–3)(1) + (–3)(x)
=
–3
– 3x
(y – 5)(–2)
= (y)(–2) + (–5)(–2)
= –2y
+
10
– (7 – 3x)
= (–1)(7) + (–1)(–3x)
= –7 +
3x
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)
6) a(c + d)
4y - 28
ac + ad
2) 3(b + 4)
7) - (-3 - r)
3b + 12
3+r
3) -5(9 - m)
8) 4(x - 8)
-45 + 5m
4x - 32
4) 5(4 - a)
9) - (2m + 3)
20 - 5a
-2m - 3
5) (7 - k)6
10) (6 - 2y) -3y
42 - 6k
6 - 2y -3y
MENTAL MATH CALCULATIONS
You are shopping for CDs.
You want to buy six CDs
for $11.95 each.
6(11.95) = 6(12 – 0.05)
= 6(12) – 6(0.05)
= 72 – 0.30
= 71.70
Use the distributive property
to calculate the total cost
mentally.
The mental math is easier if you
think of $11.95 as $12.00 – $.05.
Write 11.95 as a difference.
Use the distributive property.
Find the products mentally.
Find the difference mentally.
The total cost of 6 CDs at $11.95 each is $71.70.
SIMPLIFYING BY COMBINING LIKE TERMS
(8 + 3)x = 8x + 3x
= 11x
4x2 + 2 – x2 = 4x2 – x2 + 2
= 3x2 + 2
Use the distributive property.
Add coefficients.
Group like terms.
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 Designate one sign in front of 2x
Subtracting a Quantity
1) -(x + 6)
-x - 6
2) -(2x - 8)
-2x + 8
3) 10- (4m + 3)
10 - 4m - 3
- 4m + 7
4) 2(x - 5) - (x - 3)
2x - 10 - x + 3
x-7
5) -(3a + 1)
-3a - 1
2
6) -(-3x + 2x -7)
2
+3x - 2x + 7
7) -12 - (3y - 8)
-12 - 3y + 8
- 3y - 4
8) 4(3k - 5) - (2k + 9)
12k - 20 - 2k - 9
10k - 29
Geometric Model for Area
3 +
7
4 4(3)
4(7)
Two ways to find the total area.
Width by total length
(Order of Operations)
Sum of smaller rectangles
(Distributive Property)
4(3 + 7)
=
4 (10)
40
=
=
4(3) + 4(7)
12
+ 28
40
Geometric Model for Distributive Property
4
x
9
Two ways to find the total area.
Width by total length
9(4 + x)
Sum of smaller rectangles
=
9(4) + 9(x)