Distributive Property

What Is the Distributive Property?

The distributive property is one of the most useful tools in algebra. It tells you how to handle multiplication when you have parentheses involved. Specifically, it says that multiplying a number by a sum is the same as multiplying the number by each term separately and then adding the results.

Here's the formal statement:

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

In words: when you multiply \(a\) by the sum \(b + c\), you can "distribute" the \(a\) to both \(b\) and \(c\).

The property also works with subtraction:

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

This property connects multiplication and addition (or subtraction) in a way that makes simplifying expressions much easier. Think of it like handing out items: if you need to give 3 candies to each of 4 people and 5 people, you could count everyone first (9 people) and give out \(3 \times 9 = 27\) candies. Or you could give 3 candies to the first group (12 candies) and 3 to the second group (15 candies), for a total of 27. Same result, different approach.

How It Works

Let's start with numbers to see why this makes sense.

Example 1: Calculate \(3(4 + 5)\).

You could do this two ways:

Method 1 (Order of Operations): Add first, then multiply. $$3(4 + 5) = 3(9) = 27$$

Method 2 (Distributive Property): Distribute the 3 to both numbers inside. $$3(4 + 5) = 3 \cdot 4 + 3 \cdot 5 = 12 + 15 = 27$$

Both methods give you 27. The distributive property gives you a choice in how to approach the problem.

Example 2: Calculate \(5(10 - 2)\).

Method 1: $$5(10 - 2) = 5(8) = 40$$

Method 2 (distribute): $$5(10 - 2) = 5 \cdot 10 - 5 \cdot 2 = 50 - 10 = 40$$

Again, same answer either way.

Using It With Variables

The distributive property becomes essential when you're working with variables, because often you can't simplify what's inside the parentheses.

Example 3: Simplify \(4(x + 3)\).

You can't add \(x + 3\) because one is a variable and one is a number. But you can distribute the 4:

$$4(x + 3) = 4 \cdot x + 4 \cdot 3 = 4x + 12$$

Example 4: Simplify \(7(2y - 5)\).

Distribute the 7 to both terms: $$7(2y - 5) = 7 \cdot 2y - 7 \cdot 5 = 14y - 35$$

Example 5: Simplify \(6(3a + 4b)\).

Distribute the 6: $$6(3a + 4b) = 6 \cdot 3a + 6 \cdot 4b = 18a + 24b$$

Notice that you multiply the 6 by each term inside the parentheses. Don't skip any terms.

Example 6: Simplify \(x(x + 5)\).

Yes, you can distribute a variable too: $$x(x + 5) = x \cdot x + x \cdot 5 = x^2 + 5x$$

When you multiply \(x \times x\), you get \(x^2\). This comes up frequently when multiplying polynomials or expanding expressions with variables.

Example 6b: Simplify \(3x(2x - 4)\).

Distribute the \(3x\) to both terms: $$3x(2x - 4) = 3x \cdot 2x - 3x \cdot 4 = 6x^2 - 12x$$

Remember that \(3x \times 2x = 6x^2\) (multiply the coefficients and add the exponents).

Distributing Negative Numbers

This is where students often make mistakes, so pay close attention. When you distribute a negative number, it affects all the terms inside the parentheses.

Example 7: Simplify \(-3(x + 4)\).

Distribute the \(-3\): $$-3(x + 4) = -3 \cdot x + (-3) \cdot 4 = -3x - 12$$

Both terms become negative.

Example 8: Simplify \(-2(5y - 3)\).

Distribute the \(-2\): $$-2(5y - 3) = -2 \cdot 5y - (-2) \cdot 3 = -10y + 6$$

Notice that \(-2 \times -3 = +6\). The negative times negative gives you positive.

Example 9: Simplify \(-(x - 7)\).

When you see a negative sign in front of parentheses with no number, it's really \(-1\): $$-(x - 7) = -1(x - 7) = -1 \cdot x - (-1) \cdot 7 = -x + 7$$

This one catches a lot of people. The minus sign distributes to everything inside.

Reverse: Factoring

The distributive property works both ways. Going backward—from \(ab + ac\) to \(a(b + c)\)—is called factoring. You're pulling out the common factor.

Example 10: Factor \(6x + 9\).

Both terms share a factor of 3: $$6x + 9 = 3 \cdot 2x + 3 \cdot 3 = 3(2x + 3)$$

Example 11: Factor \(10y - 15\).

Both terms are divisible by 5: $$10y - 15 = 5 \cdot 2y - 5 \cdot 3 = 5(2y - 3)$$

Example 12: Factor \(4a + 4b\).

The common factor is 4: $$4a + 4b = 4(a + b)$$

Factoring is the distributive property in reverse, and you'll use it constantly when solving equations and working with more complex expressions.

Multiple Steps

Sometimes you need to distribute and then combine like terms.

Example 13: Simplify \(3(x + 2) + 5x\).

First, distribute the 3: $$3(x + 2) + 5x = 3x + 6 + 5x$$

Now combine like terms: $$= 8x + 6$$

Example 14: Simplify \(4(2a - 3) - 2(a + 1)\).

Distribute both: $$4(2a - 3) - 2(a + 1) = 8a - 12 - 2a - 2$$

Combine like terms: $$= 6a - 14$$

Watch that second distribution carefully. You're distributing \(-2\), so it becomes \(-2a - 2\).

Example 15: Simplify \(2(x - 4) + 3(x + 1)\).

Distribute both: $$2(x - 4) + 3(x + 1) = 2x - 8 + 3x + 3$$

Combine like terms: $$= 5x - 5$$

Practice Problems

Try these on your own, then check your answers below.

1. Simplify \(5(x + 6)\)

2. Simplify \(8(3y - 4)\)

3. Simplify \(-4(m + 7)\)

4. Simplify \(-3(2a - 5)\)

5. Simplify \(-(6 - x)\)

6. Factor \(12x + 18\)

7. Simplify \(7(n + 2) + 3n\)

8. Simplify \(5(2x - 1) - 3(x + 4)\)

Check Your Answers

1. \(5x + 30\)
   Distribute: \(5 \cdot x + 5 \cdot 6\)

2. \(24y - 32\)
   Distribute: \(8 \cdot 3y - 8 \cdot 4\)

3. \(-4m - 28\)
   Distribute: \(-4 \cdot m + (-4) \cdot 7\)

4. \(-6a + 15\)
   Distribute: \(-3 \cdot 2a - (-3) \cdot 5 = -6a + 15\)

5. \(-6 + x\) or \(x - 6\)
   Distribute the hidden \(-1\): \(-1 \cdot 6 - (-1) \cdot x = -6 + x\)

6. \(6(2x + 3)\)
   Both divisible by 6: \(6 \cdot 2x + 6 \cdot 3\)

7. \(10n + 14\)
   Distribute: \(7n + 14 + 3n\). Combine: \(10n + 14\)

8. \(7x - 17\)
   Distribute: \(10x - 5 - 3x - 12\). Combine: \(7x - 17\)

Why This Matters

The distributive property shows up everywhere in algebra. You'll use it when simplifying expressions, solving equations, multiplying polynomials, factoring, and working with formulas. It's one of those fundamental skills that makes everything else easier.

Once you master distributing, many algebra problems become much more manageable. The key is practice—work through enough examples that it becomes automatic. Make sure you're distributing to every term inside the parentheses. If you have something like \(3(x + y + 2)\), that 3 needs to multiply all three terms: \(3x + 3y + 6\). Missing even one term will throw off your entire answer.

Negative signs are where most mistakes happen. When you're distributing \(-2(x - 5)\), both terms inside get affected by that negative. You end up with \(-2x + 10\), not \(-2x - 10\). That second term becomes positive because you're multiplying \(-2)\) times \(-5\). Similarly, if you see something like \(5 - 2(x + 3)\), think of it as \(5 + (-2)(x + 3)\), which becomes \(5 - 2x - 6\) or \(-2x - 1\).

Remember that factoring is just the distributive property working backward. If you can distribute \(a(b + c)\) to get \(ab + ac\), you can also reverse the process and go from \(ab + ac\) back to \(a(b + c)\). This reverse process is incredibly useful when you're trying to simplify expressions or solve more complex equations.