Distributive Property Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Distributive Property.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The rule that multiplying a sum equals the sum of individual products: a(b+c)=ab+aca(b+c) = ab + ac. It links multiplication and addition, allowing you to break apart or combine terms.

Three packs of (2 red + 4 blue) = (3×23 \times 2 red) + (3×43 \times 4 blue) = 6 red + 12 blue.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Distributing multiplies the outside factor by each term in the parentheses, then adds the products.

Common stuck point: The procedure for distributive property is the easy part; the trap is multiplying only the first term in the parentheses. Asking "Is a single factor multiplying a sum where it must reach every term inside?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is a single factor multiplying a sum where it must reach every term inside?

Worked Examples

Example 1

easy
Expand 4(x+7)4(x + 7).

Answer

4x+284x + 28

First step

1
Identify the two terms inside the parentheses: xx and 77.

Full solution

  1. 2
    Distribute 4 to each term inside the parentheses: 4⋅x+4⋅74 \cdot x + 4 \cdot 7.
  2. 3
    Simplify: 4x+284x + 28.
The distributive property states a(b+c)=ab+aca(b + c) = ab + ac. Multiply the outside factor by each term inside the parentheses.

Example 2

medium
Expand and simplify 3(2x−5)+2(x+4)3(2x - 5) + 2(x + 4).

Example 3

easy
Expand a(b+c+d)a(b + c + d).

Example 4

medium
Show that 7×99=6937 \times 99 = 693 using the distributive property.

Example 5

hard
A theater charges $12 per adult ticket and $8 per child ticket. A group buys aa adult and cc child tickets. Write a single expression and an equivalent expanded form for the total cost.

Example 6

challenge
Prove that if a,b,ca, b, c are integers and aa divides both bb and cc, then aa divides b+cb + c.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

medium
Expand −2(3a−4b+1)-2(3a - 4b + 1).

Example 2

easy
Use the distributive property to compute 6×486 \times 48 mentally.

Example 3

easy
Expand 3(x+4)3(x + 4).

Example 4

easy
Expand 5(a+2)5(a + 2).

Example 5

easy
Expand 2(3x+5)2(3x + 5).

Example 6

easy
Expand −(a+b)-(a + b).

Example 7

easy
Use distribution to compute 6×126 \times 12 by splitting 12.

Example 8

easy
Expand 4(x−3)4(x - 3).

Example 9

easy
Factor 6x+96x + 9 by pulling out the common factor.

Example 10

easy
Expand x(x+5)x(x + 5).

Example 11

medium
Expand and simplify 3(2x+1)+4(x−2)3(2x + 1) + 4(x - 2).

Example 12

medium
Solve 2(x+3)=162(x + 3) = 16.

Example 13

medium
Expand (x+2)(x+3)(x + 2)(x + 3).

Example 14

medium
Compute 7×987 \times 98 using distribution.

Example 15

medium
Factor completely: 12x+18y−2412x + 18y - 24.

Example 16

medium
Expand −2(3x−4)-2(3x - 4).

Example 17

medium
Expand 5(2a+3b−c)5(2a + 3b - c).

Example 18

medium
Expand −3(2x−5)-3(2x - 5).

Example 19

medium
Simplify 2(x+1)+3(x+1)2(x + 1) + 3(x + 1) by factoring the common binomial.

Example 20

challenge
Expand and simplify (x+3)2(x + 3)^2.

Example 21

challenge
Use the distributive property to show a(b−c)=ab−aca(b - c) = ab - ac follows from a(b+c)=ab+aca(b+c)=ab+ac.

Example 22

challenge
Factor x2+5xx^2 + 5x and then evaluate the factored form at x=7x = 7.

Example 23

easy
Expand 7(x+2)7(x + 2).

Example 24

easy
Expand 6(y−4)6(y - 4).

Example 25

easy
Use distribution to compute 9×219 \times 21.

Example 26

easy
Factor 4x+84x + 8 by pulling out the greatest common factor.

Example 27

easy
Expand 2(5x+3y)2(5x + 3y).

Example 28

easy
Factor 9x−129x - 12.

Example 29

medium
Expand and simplify 4(2x−3)−2(x−5)4(2x - 3) - 2(x - 5).

Example 30

medium
Solve 3(x−4)=213(x - 4) = 21.

Example 31

medium
Expand (x+4)(x−3)(x + 4)(x - 3).

Example 32

medium
Use distribution to compute 25×3225 \times 32 mentally.

Example 33

medium
Factor completely: 15x2+10x15x^2 + 10x.

Example 34

medium
Expand −4(x−2y+3)-4(x - 2y + 3).

Example 35

medium
Solve 5(2x+1)−3(x−4)=325(2x + 1) - 3(x - 4) = 32.

Example 36

medium
Expand (2x−1)(x+5)(2x - 1)(x + 5).

Example 37

medium
Simplify 3(x+2y)−2(x−y)3(x + 2y) - 2(x - y).

Example 38

hard
Expand (x+1)(x2−x+1)(x + 1)(x^2 - x + 1).

Example 39

hard
Solve for xx: 4(x−1)+3(x+2)=5x+104(x - 1) + 3(x + 2) = 5x + 10.

Example 40

hard
Factor by grouping: x3+2x2+3x+6x^3 + 2x^2 + 3x + 6.

Example 41

hard
Expand (x+y+z)2(x + y + z)^2.

Example 42

challenge
Expand (a+b)(c+d)(e+f)(a + b)(c + d)(e + f).
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Background Knowledge

These ideas may be useful before you work through the harder examples.

multiplicationaddition