Properties of Operations

TipLearning Objectives

By the end of this lesson, you’ll be able to:

  • Identify and use key algebraic properties (commutative, associative, distributive).
  • Rewrite and simplify expressions using these properties.
  • Recognize identity and inverse elements for addition and multiplication.
  • Avoid common mistakes involving incorrect distribution or regrouping.

Key Ideas

Algebra relies on core properties that allow you to rewrite expressions.
These show up constantly when simplifying, factoring, expanding, or solving equations.

Commutative Property

Order can change (addition/multiplication only):

  • Addition: \(a + b = b + a\)
  • Multiplication: \(ab = ba\)

Associative Property

Grouping can change (addition/multiplication):

  • Addition: \((a + b) + c = a + (b + c)\)
  • Multiplication: \((ab)c = a(bc)\)
Property What It Means Addition Example Multiplication Example
Commutative Order can change without changing the result. \(a + b = b + a\) \(ab = ba\)
Associative Grouping can change without changing the result. \((a + b) + c = a + (b + c)\) \((ab)c = a(bc)\)

Distributive Property

Multiplication distributes over addition or subtraction:

\[ a(b + c) = ab + ac \]

And in reverse (factoring):

\[ ab + ac = a(b + c) \]

Identity Elements

  • Additive identity: \(a + 0 = a\)
  • Multiplicative identity: \(a \cdot 1 = a\)

Inverse Elements

  • Additive inverse: \(a + (-a) = 0\)
  • Multiplicative inverse: \(a \cdot \frac{1}{a} = 1\) (for \(a \ne 0\))

Zero Property

Multiplying by 0 always gives 0:

\[ a \cdot 0 = 0 \]

Important

The distributive property does not apply to exponents or absolute value.
Common incorrect statements:

  • \((a + b)^2 \ne a^2 + b^2\)
  • \(|a + b| \ne |a| + |b|\)

These mistakes often come from assuming distribution works everywhere.

Common Problem Types

1. Simplifying Expressions

Example:
\[ 3(x + 4) = 3x + 12 \]

2. Factoring Using the Distributive Property

Example:
\[ 6x + 15 = 3(2x + 5) \]

3. Regrouping Terms (Associative Property)

Example:
\[ (2 + 7) + 13 = 2 + (7 + 13) \]

4. Reordering Terms (Commutative Property)

Example:
\[ 4y + 7x = 7x + 4y \]

5. Avoiding Incorrect Distribution

Incorrect:
\[ (x + 3)^2 = x^2 + 9 \]

Correct:
\[ (x + 3)^2 = x^2 + 6x + 9 \]

Strategies

  • When expanding expressions → distribute first, then combine like terms.
  • When factoring → look for the greatest common factor.
  • Use commutative/associative properties to reorder terms for easier combining.
  • Remember: exponents do not distribute over addition or subtraction.

Worked Examples

Example 1

Question: Rewrite \(5x + 20\) in factored form.

Step-by-step Solution:

  1. Identify the greatest common factor: GCF = 5
  2. Factor it out:
    \(5x + 20 = 5(x + 4)\)

Answer: \(5(x + 4)\)

Example 2

Question: Expand \(2(3x - 5)\).

Step-by-step Solution:

  1. Distribute: \(2 \cdot 3x = 6x\)
  2. Distribute again: \(2 \cdot (-5) = -10\)

Answer: \(6x - 10\)

Example 3

Question: Simplify using regrouping:
\[ (8 + 12) + 5 \]

Step-by-step Solution:

  1. Regroup: \(8 + (12 + 5)\)
  2. Add: \(12 + 5 = 17\)
  3. Add again: \(8 + 17 = 25\)

Answer: \(25\)

Example 4

Question: Simplify: \(-3(x - 4)\).

Step-by-step Solution:

  1. Distribute \(-3\):
    \(-3 \cdot x = -3x\)
  2. Distribute again:
    \(-3 \cdot (-4) = 12\)

Answer: \(-3x + 12\)

WarningCommon Mistakes
  • Distributing exponents incorrectly: \((a+b)^2 \ne a^2 + 2ab + b^2\).
  • Forgetting to factor out the greatest common factor when factoring.
  • Misreading signs when distributing over subtraction.
  • Assuming addition/subtraction work like multiplication/division.
  • Dropping negative signs while expanding.

Practice Problems

  1. Expand: \(4(x + 7)\)
  2. Factor completely: \(12x + 18\)
  3. True or false: \((a + b)^2 = a^2 + 2ab + b^2\)
  4. Expand: \(-2(3x - 1)\)
  5. Simplify: \((5 + 9) + 3\)

1. Expand: \(4(x + 7)\)
\(4x + 28\)

2. Factor completely: \(12x + 18\)
GCF = 6 → \(6(2x + 3)\)

3. True or false: \((a + b)^2 = a^2 + 2ab + b^2\)
True.

4. Expand: \(-2(3x - 1)\)
\(-6x + 2\)

5. Simplify: \((5 + 9) + 3\)
\(14 + 3 = 17\)

Summary

  • Use commutative and associative properties to reorder or regroup expressions.
  • Use the distributive property to expand or factor.
  • Identity and inverse properties help in simplifying and solving.
  • Distribution does not apply to exponents or absolute value.
  • When factoring, look for the greatest common factor first.
  • Track signs carefully when distributing negative numbers.
  • Use regrouping to simplify mental arithmetic or rearrange expressions neatly.