Distributive Property — Expand and Factor with Examples

By MathPal Editorial Team·Published Apr 17, 2026

Grade: 6-7 | Topic: Algebra

What You Will Learn

The distributive property is one of the most-used rules in all of algebra. It lets you remove parentheses from an expression by multiplying, and it lets you factor common pieces out. This guide covers expanding brackets with positive and negative numbers, factoring out the greatest common factor, and handling expressions with multiple sets of parentheses.

Theory

The core rule

The distributive property says that multiplying a number by a group of terms added together is the same as multiplying the number by each term separately and then adding:

$a(b + c) = ab + ac$

This works because multiplication "distributes" over addition. Think of it like handing out — the $a$ gets handed to every term inside the parentheses.

Concrete example: Suppose you buy 3 packs, each containing 4 pencils and 2 erasers.

$3(4 + 2) = 3 \times 4 + 3 \times 2 = 12 + 6 = 18$

You get the same answer either way: $3 \times 6 = 18$.

With subtraction

Subtraction is just adding a negative, so the property works the same way:

$a(b - c) = ab - ac$

With negative factors

When the factor outside the parentheses is negative, remember that a negative times a positive is negative, and a negative times a negative is positive:

$-2(x + 5) = -2 \cdot x + (-2) \cdot 5 = -2x - 10$

$-3(4 - y) = -3 \cdot 4 + (-3)(-y) = -12 + 3y$

Factoring — the reverse direction

Factoring means rewriting an expression by pulling out a common factor. It is the distributive property used in reverse:

$ab + ac = a(b + c)$

To factor an expression:

1. Find the greatest common factor (GCF) of all terms.
2. Divide each term by the GCF.
3. Write the GCF outside and the divided terms inside parentheses.

For example, factor $10x + 15$:

• GCF of $10$ and $15$ is $5$.
• $10x \div 5 = 2x$ and $15 \div 5 = 3$.
• Result: $5(2x + 3)$.

Worked Examples

Example 1 — Basic expansion

Expand $4(x + 3)$.

Step 1: Multiply $4$ by $x$: $4x$.

Step 2: Multiply $4$ by $3$: $12$.

Step 3: Combine: $4x + 12$.

Example 2 — Negative factor

Expand $-5(2y - 7)$.

Step 1: $-5 \times 2y = -10y$.

Step 2: $-5 \times (-7) = 35$ (negative times negative is positive).

Step 3: Combine: $-10y + 35$.

Example 3 — Two sets of brackets, then combine

Simplify $3(x + 2) + 2(x - 4)$.

Step 1: Distribute the first bracket: $3x + 6$.

Step 2: Distribute the second bracket: $2x - 8$.

Step 3: Combine all terms: $3x + 6 + 2x - 8$.

Step 4: Combine like terms: $(3x + 2x) + (6 - 8) = 5x - 2$.

Example 4 — Factoring with variables

Factor $12a + 18ab$.

Step 1: Find the GCF of $12a$ and $18ab$.

• GCF of $12$ and $18$ is $6$.
• Both terms contain at least one $a$.
• GCF is $6a$.

Step 2: Divide each term: $12a \div 6a = 2$ and $18ab \div 6a = 3b$.

Step 3: Write the result: $6a(2 + 3b)$.

Check: $6a \times 2 + 6a \times 3b = 12a + 18ab$. Correct.

Common Mistakes

Mistake 1 — Forgetting to distribute to every term

❌ $3(x + 4) = 3x + 4$

✅ $3(x + 4) = 3x + 12$ — the $3$ must multiply both $x$ and $4$.

Mistake 2 — Sign errors with a negative factor

❌ $-2(x - 5) = -2x - 10$

✅ $-2(x - 5) = -2x + 10$ — negative times negative $5$ gives positive $10$.

Mistake 3 — Not finding the greatest common factor

❌ Factor $8x + 12$: $2(4x + 6)$

✅ $4(2x + 3)$ — while $2(4x + 6)$ is technically correct, you should always factor out the GCF, which is $4$ here, not just $2$.

Practice Problems

Problem 1: Expand $5(y + 6)$.

Show Answer

$5y + 30$

Problem 2: Expand $-3(2x + 1)$.

Show Answer

$-6x - 3$

Problem 3: Simplify $4(a - 3) + 2(a + 5)$.

Show Answer

Distribute: $4a - 12 + 2a + 10$.

Combine: $6a - 2$.

Problem 4: Factor $15m - 20$.

Show Answer

GCF of $15$ and $20$ is $5$.

$15m - 20 = 5(3m - 4)$

Problem 5: Factor $9x^{2} + 6x$.

Show Answer

GCF of $9x^{2}$ and $6x$ is $3x$.

$9x^{2} + 6x = 3x(3x + 2)$

Summary

• The distributive property states $a(b + c) = ab + ac$ — multiply the outside factor by every term inside.
• It works with subtraction and negative numbers — just be careful with signs.
• Factoring is the reverse: find the GCF of all terms and write it outside parentheses.
• After distributing, always combine like terms to finish simplifying.
• This property is the gateway to solving equations, working with polynomials, and simplifying complex expressions.

Related Topics

• Combining Like Terms — simplify after distributing
• Variables and Expressions — foundational algebra vocabulary
• Solving Two-Step Equations — apply distribution to solve equations

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