Distributive Property: Rules and Examples

What is the distributive property?

The distributive property says that multiplying a number by a sum gives the same result as multiplying it by each part and adding: a(b + c) = ab + ac. You "distribute" the outside factor to every term inside the parentheses.

With numbers you can check it right away. 3(4 + 5) = 3 × 9 = 27, and 3 × 4 + 3 × 5 = 12 + 15 = 27. With variables, you can't add the terms inside first, so distributing is the only way to remove the parentheses.

A picture helps. Think of a rectangle 3 units tall and 4 + 5 units wide. You can find its area all at once, 3 × 9, or split it into a 3 by 4 piece and a 3 by 5 piece and add them. Both give 27, and that split is exactly what a(b + c) = ab + ac describes.

How do you distribute?

Multiply the term outside the parentheses by each term inside, keeping each term's sign, then write the results with their signs. Combine like terms afterward if there are any.

1. Start with 3(2x + 5).
2. Multiply 3 by 2x to get 6x.
3. Multiply 3 by 5 to get 15.
4. Write the result: 6x + 15.

How does the distributive property help with mental math?

Split one of the numbers into an easy sum or difference, then distribute. It turns a hard product into two easy ones.

For 7 × 98, think of 98 as 100 − 2. Then 7(100 − 2) = 700 − 14 = 686. For 6 × 45, use 40 + 5: 6(40 + 5) = 240 + 30 = 270.

It works for bigger numbers too. 15 × 12 is 15(10 + 2) = 150 + 30 = 180. Pick the split that gives you a round number, usually a multiple of 10 or 100.

Subtraction splits are often easier than addition ones. For 9 × 199, use 9(200 − 1) = 1800 − 9 = 1791.

How do you use the distributive property with negatives?

Multiply the negative number by every term, and every sign inside flips. A negative times a positive is negative, and a negative times a negative is positive.

For −4(x − 3), you get −4 × x = −4x and −4 × (−3) = +12, so the answer is −4x + 12. A minus sign alone in front of parentheses means −1, so −(3x − 2) = −3x + 2.

It helps to say the sign with each product as you write it: negative four times x is −4x, and negative four times negative three is +12. Saying it out loud slows you down just enough to catch a flipped sign.

Does the distributive property work over subtraction?

Yes. a(b − c) = ab − ac, because subtracting c is the same as adding −c. So 8(x − 5) = 8x − 40.

Subtraction between two groups is where it gets tricky. In 2(x + 4) − 3(x − 1), the second group is multiplied by −3, not 3. That gives 2x + 8 − 3x + 3, which simplifies to −x + 11.

A good habit is to rewrite the subtraction as adding a negative before you distribute. Then 2(x + 4) − 3(x − 1) becomes 2(x + 4) + (−3)(x − 1), and it's clear that −3 multiplies both the x and the −1.

How do you use the distributive property to solve equations?

Distribute first to clear the parentheses, then solve as usual. For 3(x − 2) = 12, distributing gives 3x − 6 = 12.

How do you distribute two binomials?

Distribute each term of the first binomial across the whole second binomial, then combine like terms. FOIL is just a name for this when both factors have two terms.

For (x + 2)(x + 5): x(x + 5) + 2(x + 5) = x² + 5x + 2x + 10 = x² + 7x + 10. The FOIL calculator shows the four products labeled one by one.

Signs carry through the same way. For (2x − 3)(x + 4), distribute 2x to get 2x² + 8x, then distribute −3 to get −3x − 12. Together that is 2x² + 5x − 12.

How is factoring related to the distributive property?

Factoring out a common factor is the distributive property in reverse. Instead of multiplying a factor in, you find the factor every term shares and pull it out.

For 12x + 18, the greatest common factor is 6. Divide each term by 6 to get 2x and 3, so 12x + 18 = 6(2x + 3). Distributing 6 back in returns 12x + 18, which is how you check it.

The common factor can include a variable or a negative. In 5x² − 15x, both terms share 5x, so the answer is 5x(x − 3). For −6x − 9, pulling out −3 leaves positive terms inside: −3(2x + 3). The how to find GCF article walks through finding the largest factor when the numbers are bigger.

Does the distributive property work with fractions and longer expressions?

Yes. The outside factor can be a fraction or a decimal, and the parentheses can hold any number of terms. You still multiply the outside factor by every term.

With a fraction: (1/2)(4x − 6) = 2x − 3, because half of 4x is 2x and half of −6 is −3. With three terms: 3(x² − 2x + 4) = 3x² − 6x + 12. Count the terms inside before you start, and make sure you end with the same number of terms.

What are common distributive property mistakes?

These come up over and over in homework, and each one is easy to fix once you spot it.

1. Multiplying only the first term: 3(x + 4) is 3x + 12, not 3x + 4.
2. Missing the sign on the second term: −4(x − 3) is −4x + 12, not −4x − 12.
3. Treating a subtraction between groups as only applying to the first term inside.
4. Distributing over multiplication. 2(3x) is 6x. There is only one term inside, so there is nothing to spread out.
5. Writing (x + 3)² as x² + 9. It is (x + 3)(x + 3) = x² + 6x + 9.