The Distributive Property

The distributive property is one of the most important tools in algebra. It connects multiplication and addition, letting you remove parentheses and simplify expressions. Every time you expand an expression like $3(x + 5)$, you are using this property.

The Rule

The distributive property states:

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

In words: multiply the number outside the parentheses by each term inside, then add the results.

The property also works with subtraction:

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

And it works when the factor is on the right side:

$(b + c) \cdot a = ba + ca$

Distributing with Numbers

Before using variables, let’s confirm the property works with plain numbers.

Example 1: Verify $4(3 + 5)$ using distribution

Method 1 — Parentheses first:

$4(3 + 5) = 4(8) = 32$

Method 2 — Distribute:

$4(3 + 5) = 4 \cdot 3 + 4 \cdot 5 = 12 + 20 = 32$

Both methods give 32. The distributive property is valid.

Example 2: Mental math with $6 \times 98$

Rewrite $98$ as $100 - 2$:

$6 \times 98 = 6(100 - 2) = 600 - 12 = 588$

This is a powerful mental math shortcut. Instead of multiplying $6 \times 98$ directly, you compute two simpler products.

Example 3: Mental math with $7 \times 53$

$7 \times 53 = 7(50 + 3) = 350 + 21 = 371$

Distributing with Variables

Example 4: Expand $3(x + 7)$

Multiply $3$ by each term inside the parentheses:

$3(x + 7) = 3 \cdot x + 3 \cdot 7 = 3x + 21$

Answer: $3x + 21$

Example 5: Expand $5(2y - 4)$

$5(2y - 4) = 5 \cdot 2y - 5 \cdot 4 = 10y - 20$

Answer: $10y - 20$

Example 6: Expand $-2(3a + 1)$

When distributing a negative number, be careful with signs:

$-2(3a + 1) = (-2)(3a) + (-2)(1) = -6a + (-2) = -6a - 2$

Answer: $-6a - 2$

Example 7: Expand $-(x - 9)$

A lone negative sign in front of parentheses means $-1$ times each term:

$-(x - 9) = (-1)(x) + (-1)(-9) = -x + 9$

Answer: $-x + 9$

Notice that both signs flipped: $x$ became $-x$, and $-9$ became $+9$.

Distributing and Then Combining Like Terms

In many problems, distribution is just the first step. After expanding, you combine like terms to fully simplify.

Example 8: Simplify $2(x + 3) + 4x$

Step 1 — Distribute:

$2x + 6 + 4x$

Step 2 — Combine like terms:

$6x + 6$

Answer: $6x + 6$

Example 9: Simplify $3(2a - 1) - 2(a + 5)$

Step 1 — Distribute both sets of parentheses:

$6a - 3 - 2a - 10$

Note the second distribution: $-2(a + 5) = -2a - 10$ (the negative distributes to both terms).

Step 2 — Combine like terms:

$4a - 13$

Answer: $4a - 13$

Example 10: Simplify $4(x + 2) + 3(x - 5)$

Step 1 — Distribute:

$4x + 8 + 3x - 15$

Step 2 — Combine:

$7x - 7$

Answer: $7x - 7$

Distributing a Variable

The distributive property works the same way when a variable is the factor outside.

Example 11: Expand $x(x + 6)$

$x(x + 6) = x \cdot x + x \cdot 6 = x^2 + 6x$

Answer: $x^2 + 6x$

Real-World Application: Retail — Discounted Pricing

A store runs a promotion: every item is $5 off the regular price. A customer buys 4 items with regular prices of $12, $18, $9, and $15. The total with the discount is:

$(12 - 5) + (18 - 5) + (9 - 5) + (15 - 5)$

Using the distributive property in reverse (factoring), this is equivalent to:

$(12 + 18 + 9 + 15) - 4 \times 5 = 54 - 20 = 34$

Answer: The discounted total is $34. A cashier who recognizes this pattern can compute totals faster — add all regular prices, then subtract the number of items times the discount.

Real-World Application: Carpentry — Perimeter Calculation

A carpenter needs to calculate the perimeter of a rectangular deck that is $(2x + 3)$ feet long and $x$ feet wide. The perimeter formula is:

$P = 2(\text{length} + \text{width})$

$P = 2((2x + 3) + x) = 2(3x + 3)$

Distribute:

$P = 6x + 6$

If $x = 4$ feet:

$P = 6(4) + 6 = 24 + 6 = 30 \text{ feet}$

The carpenter needs 30 feet of decking border material. Simplifying with the distributive property first made the substitution easy.

Common Mistakes to Avoid

1. Distributing to only the first term. In $3(x + 5)$, you must multiply $3$ by both $x$ AND $5$: $3(x + 5) = 3x + 15 \quad \text{(not } 3x + 5\text{)}$

2. Forgetting to distribute the negative. In $-2(x + 4)$, the $-2$ multiplies both terms: $-2(x + 4) = -2x - 8 \quad \text{(not } -2x + 4\text{)}$

3. Distributing across multiplication instead of addition. The distributive property links multiplication with addition or subtraction. It does NOT apply to $3(x \cdot y)$ — that is just $3xy$.

4. Forgetting to combine like terms after distributing. Distribution is often the first step of simplification. Always check whether like terms can be combined afterward.

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• The distributive property states that $a(b + c) = ab + ac$ — multiply the outside factor by every term inside
• It works with subtraction too: $a(b - c) = ab - ac$
• Distributing a negative flips the sign of every term inside the parentheses
• After distributing, always combine like terms to finish simplifying
• The property doubles as a mental math shortcut: $7 \times 98 = 7(100 - 2) = 700 - 14 = 686$
• In applied settings, distribution helps simplify formulas before substituting actual measurements

Return to Pre-Algebra for more topics in this section.