How do you distribute?
Multiply the outside term by each term inside the brackets, one at a time, and write the results with their signs. The brackets then disappear.
For 4(x + 3), multiply 4 by x and 4 by 3 to get 4x + 12. With subtraction inside, the sign carries through: 5(2x − 1) = 10x − 5.
You can check a distribution with any number. At x = 5, 4(x + 3) is 4 × 8 = 32, and 4x + 12 is 20 + 12 = 32. When both forms give the same value, they are the same expression written two ways.
How do you distribute a variable?
Multiply the coefficients and add the exponents of the variable. 3x times x² is 3x³, because x × x² = x¹⁺² = x³.
Each term's power goes up by one, since the outside term has one x. Nothing combines afterward because the three powers are all different.
Watch the middle term in particular. The inside term is −2x, so the product is 3x × −2x = −6x². Students often drop the sign or forget to multiply the x parts, which gives −6x instead.
How do you distribute a negative number?
Multiply the negative by every term inside, and every sign inside flips. −3(2x − 5) becomes −6x + 15, because −3 × −5 = +15.
Here is a full example where you distribute and then combine like terms.
A lone minus sign in front of brackets counts as −1. So 5x − (2x − 3) means 5x − 2x + 3, which is 3x + 3. Forgetting to flip the last sign is the most common distributive property mistake.
A quick rule for signs: a negative times a positive gives a negative, and a negative times a negative gives a positive. Say each sign out loud as you multiply, and write it down before the number.
Distributive property examples
The table shows a range of inputs, from a plain number outside to a variable and a negative. Each result was checked with the calculator.
| Expression | After distributing |
|---|---|
| 4(x + 3) | 4x + 12 |
| −3(2x − 5) | −6x + 15 |
| 3x(x² − 2x + 5) | 3x³ − 6x² + 15x |
| 2(3x − 4) − (x − 5) | 5x − 3 |
| 2x(3x + 4) − x² | 5x² + 8x |
How do you distribute two brackets in one expression?
Distribute each bracket separately, then combine like terms across the whole expression. Treat the sign in front of each bracket as part of the number you distribute.
The second bracket has no number in front, so it is multiplied by −1. That turns −5 into +5. Check with x = 2: the original gives 2(2) − (−3) = 7, and 5(2) − 3 = 7.
How is the distributive property used to solve equations?
You distribute to clear the brackets, combine like terms, and then solve as usual. It is often the first step in a multi-step equation.
In 3(x − 4) + 2x = 13, distributing gives 3x − 12 + 2x = 13. Combining gives 5x − 12 = 13, so 5x = 25 and x = 5. The linear equation solver shows the remaining steps if you want to check your work.
How do you use the distributive property backwards?
Reading ab + ac = a(b + c) from left to right is factoring out the greatest common factor. You find what every term shares and pull it outside the brackets.
In 8x + 12, both terms share 4, so 8x + 12 = 4(2x + 3). In 6x² + 9x, the terms share 3x, so the result is 3x(2x + 3). To check a factored answer, distribute it again and see if you get the original.
When the first term is negative, it is usual to factor out a negative number. −2x − 6 becomes −2(x + 3), and distributing the −2 brings back both minus signs. With larger numbers, 12x² − 8x factors as 4x(3x − 2); the GCF calculator helps when the common factor isn't obvious.
How does the distributive property help with mental math?
Split one number into an easy sum, then multiply each part. To find 7 × 103, think 7 × (100 + 3) = 700 + 21 = 721.
It works with subtraction too: 6 × 98 = 6 × (100 − 2) = 600 − 12 = 588. This is the same rule you use in algebra, just with numbers in place of x.
Shopping math works the same way. Four items at $2.99 each is 4 × (3 − 0.01), which is 12 − 0.04 = $11.96. Rounding to a friendly number and then correcting is the distributive property at work.