The Distributive Property

The Distributive Property

The distributive property lets you multiply a single term by a sum or difference inside parentheses without needing to compute the sum first. In symbols:

and similarly a(b - c) = ab - ac. In words: multiply the outside factor by each term inside the parentheses, then add or subtract the results.

Why It Works

Picture buying 3 bags, each containing 2 apples and 5 oranges. The total number of pieces of fruit is 3(2 + 5), but you could also count all the apples (3 \times 2 = 6) and all the oranges (3 \times 5 = 15) separately and add: 6 + 15 = 21. Both methods give the same answer, which is exactly what the distributive property guarantees.

Distributing a Negative Number

Distributing a negative sign flips the sign of every term inside the parentheses. This is one of the most common places students lose a sign, so slow down here:

Notice that -2 \times x = -2x, but -2 \times (-5) = 10 — a negative times a negative is a positive.

Distributing, Then Combining Like Terms

Many problems ask you to distribute first and then simplify by combining like terms. Always distribute completely before you try to combine anything.

Example

Simplify 4(2x - 3) - 3(x + 5).

1. Distribute the 4: 4(2x - 3) = 8x - 12.
2. Distribute the -3 (careful with the sign!): -3(x + 5) = -3x - 15.
3. Rewrite the full expression: 8x - 12 - 3x - 15.
4. Combine like terms: (8x - 3x) + (-12 - 15) = 5x - 27.

So the simplified expression is 5x - 27

Distributing a Fraction or a Variable

The distributive property isn't limited to whole-number factors. You can distribute a fraction, such as \tfrac{1}{2}(4x + 6) = 2x + 3, or even a variable factor, such as x(x + 3) = x^2 + 3x. The rule is identical: multiply the outside factor by every term inside the parentheses.

Reversing the Process: Factoring

Running the distributive property backward is called factoring. If you notice that every term in an expression shares a common factor, you can pull that factor back out front. For example, 6x + 9 can be rewritten as 3(2x + 3), since 3 divides evenly into both 6 and 9. Factoring will become an essential tool in later algebra courses, but recognizing the connection to distributing now will make it feel familiar later.

Key Takeaways

• a(b+c) = ab + ac; the outside factor multiplies every term inside.
• Distributing a negative number flips the sign of each term inside the parentheses.
• Distribute fully before combining like terms.
• Double-check sign errors — they are the most common mistake with this property.