Distributive Property

How do you multiply a number by a sum? The distributive property is one of the most important properties in algebra - it helps simplify expressions and solve equations!

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What Is the Distributive Property?

The distributive property says you can distribute multiplication over addition (or subtraction).

Formula: a(b + c) = ab + ac

In words: Multiply the outside number by EACH term inside the parentheses.

Example: 3(4 + 5)

Method 1: Add first 3(4 + 5) = 3(9) = 27

Method 2: Distribute 3(4 + 5) = 3(4) + 3(5) = 12 + 15 = 27

Both give the same answer!

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Why It Works

Visual example: 3(4 + 5)

Think: 3 groups of (4 + 5)

(4 + 5) + (4 + 5) + (4 + 5)

Rearrange: (4 + 4 + 4) + (5 + 5 + 5)

Which is: 3(4) + 3(5) = 12 + 15 = 27

The property lets us break apart and recombine!

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Basic Examples

Example 1: 5(2 + 3)

Distribute the 5: 5(2) + 5(3) = 10 + 15 = 25

Check: 5(5) = 25 ✓

Example 2: 7(6 + 1)

7(6) + 7(1) = 42 + 7 = 49

Example 3: 4(10 + 2)

4(10) + 4(2) = 40 + 8 = 48

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Distributive Property with Subtraction

Works the same with subtraction!

Formula: a(b - c) = ab - ac

Example: 6(8 - 3)

Distribute: 6(8) - 6(3) = 48 - 18 = 30

Check: 6(5) = 30 ✓

Important: Keep the subtraction sign with the second term!

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Step-by-Step Process

Example: Simplify 8(x + 4)

Step 1: Multiply outside number by first term 8 × x = 8x

Step 2: Multiply outside number by second term 8 × 4 = 32

Step 3: Combine with the operation (+ or -) 8x + 32

Answer: 8(x + 4) = 8x + 32

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Distributing with Variables

Example 1: 5(x + 3)

5(x) + 5(3) = 5x + 15

Example 2: 7(y - 2)

7(y) - 7(2) = 7y - 14

Example 3: 3(2n + 5)

3(2n) + 3(5) = 6n + 15

Note: When distributing to a variable term, multiply the coefficients!

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Negative Outside the Parentheses

Be careful with negative signs!

Example 1: -2(x + 5)

-2(x) + (-2)(5) = -2x - 10

Both terms become negative!

Example 2: -3(y - 4)

-3(y) - (-3)(4) = -3y + 12

Negative times negative gives positive!

Example 3: -(a + 6)

This means -1(a + 6): -1(a) + (-1)(6) = -a - 6

The negative distributes to all terms!

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More Complex Expressions

Example 1: 4(2x + 3y)

4(2x) + 4(3y) = 8x + 12y

Example 2: 5(3a - 2b + 1)

5(3a) + 5(-2b) + 5(1) = 15a - 10b + 5

Distribute to EVERY term inside!

Example 3: -2(4m - 3n + 7)

-2(4m) + (-2)(-3n) + (-2)(7) = -8m + 6n - 14

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Distributive Property with Fractions

Example 1: 1/2(6 + 4)

1/2(6) + 1/2(4) = 3 + 2 = 5

Example 2: 2/3(9x + 6)

2/3(9x) + 2/3(6) = 6x + 4

Example 3: -1/4(8y - 12)

-1/4(8y) - 1/4(-12) = -2y + 3

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Reverse: Factoring Out

The distributive property works backwards too!

Example: 6x + 9

Factor out the GCF (3): 3(2x + 3)

Check by distributing: 3(2x) + 3(3) = 6x + 9 ✓

This is called factoring!

Example 2: 12a - 8

Factor out 4: 4(3a - 2)

Check: 4(3a) - 4(2) = 12a - 8 ✓

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Mental Math with Distributive Property

Example: Calculate 7 × 98

Think: 98 = 100 - 2

7 × 98 = 7(100 - 2) = 7(100) - 7(2) = 700 - 14 = 686

Much easier than 7 × 98 directly!

Example 2: Calculate 5 × 103

5 × 103 = 5(100 + 3) = 500 + 15 = 515

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Combining Like Terms After Distributing

Example: 3(x + 2) + 4(x + 1)

Step 1: Distribute both 3x + 6 + 4x + 4

Step 2: Combine like terms (3x + 4x) + (6 + 4) 7x + 10

Answer: 7x + 10

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Using Distributive Property in Equations

Example: Solve 2(x + 3) = 14

Step 1: Distribute 2x + 6 = 14

Step 2: Subtract 6 2x = 8

Step 3: Divide by 2 x = 4

Check: 2(4 + 3) = 2(7) = 14 ✓

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Common Patterns

Pattern 1: a(x + y) = ax + ay

Pattern 2: a(x - y) = ax - ay

Pattern 3: -a(x + y) = -ax - ay

Pattern 4: -(x - y) = -x + y

Pattern 5: a(bx + c) = abx + ac

Recognizing patterns speeds up your work!

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Multiple Terms Outside

What about (2 + 3)(4 + 5)?

This uses the distributive property twice!

Method 1: Add first (2 + 3)(4 + 5) = 5 × 9 = 45

Method 2: Distribute each term 2(4 + 5) + 3(4 + 5) = 2(9) + 3(9) = 18 + 27 = 45

Note: Full FOIL method comes later in algebra!

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Real-World Applications

Shopping: 3 items at $5 each + 3 items at$2 each = 3($5 +$2) = 3($7) =$21

Or: 3($5) + 3($2) = $15 +$6 = $21

Area: Rectangle split in two parts Total area = width × (length₁ + length₂) = width × length₁ + width × length₂

Grouping: 5 groups with 3 boys and 4 girls each Total people = 5(3 + 4) = 5(7) = 35 Or: 5(3) + 5(4) = 15 + 20 = 35

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Order of Operations

Remember PEMDAS!

With parentheses: 2(3 + 4)

1. Add inside: 2(7)
2. Multiply: 14

With distributive property: 2(3 + 4)

1. Distribute: 2(3) + 2(4)
2. Multiply: 6 + 8
3. Add: 14

Same answer both ways!

Choose the easier method for the problem!

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Common Mistakes to Avoid

❌ Mistake 1: Forgetting to distribute to all terms

• Wrong: 3(x + 4) = 3x + 4
• Right: 3(x + 4) = 3x + 12

❌ Mistake 2: Not distributing negative signs

• Wrong: -2(x + 3) = -2x + 3
• Right: -2(x + 3) = -2x - 6

❌ Mistake 3: Distributing when you should add first

• Don't always need to distribute!
• 5(10) is easier than distributing 5(6 + 4)

❌ Mistake 4: Sign errors with subtraction

• Wrong: 4(x - 2) = 4x - 2
• Right: 4(x - 2) = 4x - 8

❌ Mistake 5: Forgetting to multiply coefficients

• Wrong: 3(2x) = 2x
• Right: 3(2x) = 6x

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When to Use Distributive Property

Use when:

• Variables inside parentheses: 5(x + 2)
• Simplifying expressions: 3(2a + 1) + 4
• Solving equations: 2(x + 3) = 10
• Mental math: 6 × 99

Don't need to use when:

• Simple numbers: 4(10) = 40
• Can add first easily: 5(3 + 2) = 5(5) = 25

Choose the easiest path!

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Problem-Solving Strategy

To distribute:

1. Identify what's outside parentheses
2. Multiply it by EACH term inside
3. Keep track of + and - signs
4. Simplify the result

To solve equations with parentheses:

1. Distribute first
2. Combine like terms
3. Solve using inverse operations
4. Check your answer

For mental math:

1. Break number into easier parts
2. Use distributive property
3. Calculate mentally

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Practice Patterns

a(b + c) patterns:

• 2(x + 5) = 2x + 10
• 3(y + 1) = 3y + 3
• 7(m + 4) = 7m + 28

a(b - c) patterns:

• 4(x - 2) = 4x - 8
• 5(n - 3) = 5n - 15
• 6(p - 1) = 6p - 6

Negative outside:

• -3(x + 4) = -3x - 12
• -2(y - 5) = -2y + 10
• -(a + 3) = -a - 3

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Quick Reference

Basic Formula: a(b + c) = ab + ac a(b - c) = ab - ac

With Variables: a(x + b) = ax + ab

Negative Outside: -a(b + c) = -ab - ac -a(b - c) = -ab + ac

Steps:

1. Multiply outside by first term
2. Multiply outside by each other term
3. Keep the operations (+ or -)
4. Simplify

Remember: Distribute to EVERY term!

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Practice Tips

Tip 1: Write out each step

• Don't try to do it all in your head
• Helps avoid sign errors

Tip 2: Check by substituting

• Pick a value for the variable
• Check that both sides equal

Tip 3: Watch those signs!

• Negative outside affects all terms
• Negative × Negative = Positive

Tip 4: Use it for mental math

• Break hard numbers into easier parts
• 8 × 97 = 8(100 - 3) = 800 - 24 = 776

Tip 5: Practice reverse (factoring)

• Helps understand the property deeper
• Useful for future algebra

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Summary

The distributive property:

• Multiplies outside number by each term inside
• Works with addition and subtraction
• Essential for algebra

Formula: a(b + c) = ab + ac

Key points:

• Distribute to ALL terms
• Keep track of signs (negative × negative = positive)
• Works forward (distributing) and backward (factoring)
• Useful for simplifying, solving, and mental math

Applications:

• Simplifying algebraic expressions
• Solving equations with parentheses
• Mental math tricks
• Area problems
• Real-world calculations

The distributive property is a foundation for all future algebra - master it now!