1.5: The Distributive Property
Algebra 1 (H)
Hawaii Content & Performance Standards
•
Standard 10: Patterns, Functions, and Algebra: SYMBOLIC REPRESENTATION: Use symbolic forms to represent,
model, and analyze mathematical situations
•
Benchmark MA.AI.10.3: Justify the steps used in simplifying expressions and solving equations and inequalities
Goals:
•
•
Use the Distributive Property to evaluate expressions.
Use the Distributive Property to simplify expressions.
Introduction to Distributive Property (Example)
Last time…
5 ∙ (6 + 4)
= 5 ∙ (10)
= 50
GEMDAS
THIS time…
5 ∙ (6 + 4)
= 5(6) + 5(4) Distributive Prop.
= 30 + 20
= 50
Key Concept: Distributive Property
For any numbers a, b, and c:
Examples:
Example 1: Distribute Over Addition
Rewrite 5(7 + 2) using the Distributive Property. Then evaluate.
5 (7 + 2)
Rewrite 8(10 + 4) using the Distributive Property. Then evaluate.
8 (10 + 4) = 8 (10) + 8 (4)
= 5 (7) + 5 (2)
Distributive Property
= 35
Multiply
=
Add
= 112
= 45
+ 10
80
+ 32
Distributive Property
Multiply
Add
Example 2: Distribute Over Subtraction
Rewrite (16 –7)3 using the Distributive Property. Then evaluate.
(16 –7)3
Rewrite (12 – 3)6 using the Distributive Property. Then evaluate.
= 16 (3) — 7 (3) Distributive Property
=
48
— 21
= 27
(12 – 3)6 = 12(6) — 3(6)
Multiply
=
Subtract
72
— 18
= 54
Distributive Property
Multiply
Subtract
Example 3: Use the Distributive Property
Use the Distributive Property to find each product.
a.) 15 ∙ 99
15 ∙ 99 =
15 (100—1)
Think: 99 = 100—1
=
15(100) – 15(1)
Distributive Property
=
1500 – 15
Multiply
=
1485
Subtract
1
2
b.) 35 �2 5�
1
5
1
5
35 �2 � = 35 �2 + �
1
5
= 35(2) + 35� �
=
70 + 7
=
77
1
5
Think: 2 = 2 +
1
5
Distributive Property
c.) 27 �3 3�
2
3
2
3
27 �3 � = 27 �3 + �
Multiply
Add
2
3
2
3
Think: 3 = 3 +
= 27(3) + 27� �
Distributive Property
=
81 + 18
=
99
Add
Multiply
Simplifying Expressions
Simplest Form: an expression has no like terms or parentheses.
Term: a number, variable or product/quotient of numbers and variables (separated by addition or subtraction signs.
Like Terms: Terms that contain the same variable and same exponent.
Coefficient: the number in front of a variable.
3𝑎2 + 5𝑎2 + 2𝑎
Like Terms
2
3
Unlike Terms
Example 4: Algebraic Expressions
Rewrite each product using the Distributive Property. Then simplify.
a.)
c.)
12(𝑦 + 3)
12(𝑦 + 3) = 12(y) + 12 (3) Distr. Prop.
= 12y + 36
b.)
5(𝑔 − 9) = 5(g) + 5 (-9)
Multiply
4(𝑦 2 + 8𝑦 + 2)
5(𝑔 − 9)
= 5g — 45
d.)
4(𝑦 2 + 8𝑦 + 2) = 4(𝑦 2 ) + 4(8𝑦) + 4 (2)
Distr. Prop.
Multiply
3(2𝑥 2 + 4𝑥 − 1)
3(2𝑥 2 + 4𝑥 − 1) = 3(2𝑥 2 ) + 3(4𝑥) + 3 (−1)
= 4𝑦 2 + 32𝑦 + 8
= 6𝑥 2 + 12𝑥 − 3
Example 5: Combine Like Terms
Simplify each expression.
a.)
17𝑎 + 21𝑎 = 38𝑎
c.)
15𝑥 + 18𝑥 = 33𝑥
b.)
12𝑏 2 − 8𝑏 2 + 6𝑏 = 4𝑏 2 + 6𝑏
d.)
10𝑛 + 3𝑛2 + 9𝑛2 = 10𝑛 + 12𝑛2