The Distributive Property

The distributive property allows us to multiply a number or algebraic term by every term inside parentheses.

The basic rule is:

a(b + c) = ab + ac

For subtraction:

a(b − c) = ab − ac

For example:

3(x + 4)

means that 3 multiplies both terms inside the parentheses:

3(x) + 3(4)

Therefore:

3(x + 4) = 3x + 12

The distributive property is essential for expanding expressions, simplifying algebra, solving equations, and factoring.

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Why Distribution Works

Consider:

3(x + 2)

This means three groups of:

x + 2

So we can write:

(x + 2) + (x + 2) + (x + 2)

Combine like terms:

x + x + x + 2 + 2 + 2

which gives:

3x + 6

Therefore:

3(x + 2) = 3x + 6

Distribution is not simply a rule to memorize. It follows directly from multiplication.

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An Area Model

The distributive property can also be understood using the area of a rectangle.

Suppose a rectangle has:

height = 4

and:

width = x + 3

Its area is:

A = 4(x + 3)

We could divide the rectangle into two smaller rectangles.

Their areas are:

4x

and:

4(3) = 12

So the total area is:

4x + 12

Therefore:

4(x + 3) = 4x + 12

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Expanding Expressions

To expand an expression means to remove parentheses by applying multiplication.

For example:

5(x + 2)

Distribute 5:

5(x) + 5(2)

Simplify:

5x + 10

Therefore:

5(x + 2) = 5x + 10

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A Simple Distribution Process

When you see:

a(b + c)

use:

Multiply → Multiply → Simplify

Multiply a by the first term:

ab

Multiply a by the second term:

ac

Then write:

ab + ac

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Distribution with Addition

Expand:

6(x + 3)

Multiply 6 by x:

6x

Multiply 6 by 3:

18

Therefore:

6(x + 3) = 6x + 18

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Distribution with Subtraction

Expand:

4(x − 5)

Multiply 4 by x:

4x

Multiply 4 by −5:

−20

Therefore:

4(x − 5) = 4x − 20

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The Sign Is Important

Consider:

7(x − 2)

The second term is:

−2

So:

7(−2) = −14

Therefore:

7(x − 2) = 7x − 14

Always keep the sign attached to the term.

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Distributing a Negative Number

Negative numbers require extra care.

Expand:

−3(x + 4)

Multiply:

−3(x) = −3x

and:

−3(4) = −12

Therefore:

−3(x + 4) = −3x − 12

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Negative Outside and Subtraction Inside

Expand:

−2(x − 5)

Multiply:

−2(x) = −2x

and:

−2(−5) = +10

Therefore:

−2(x − 5) = −2x + 10

Remember:

negative × negative = positive

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The Invisible −1

An expression such as:

−(x + 6)

really means:

−1(x + 6)

Distribute −1:

−x − 6

Therefore:

−(x + 6) = −x − 6

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Another Example

Expand:

−(3x − 7)

Multiply every term by −1:

−3x + 7

Therefore:

−(3x − 7) = −3x + 7

This is why removing parentheses preceded by a minus sign changes the signs of all terms inside.

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Distributing Variables

The factor outside the parentheses does not have to be a number.

Expand:

x(x + 4)

Multiply x by each term:

x(x) + x(4)

Therefore:

x² + 4x

So:

x(x + 4) = x² + 4x

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Distributing Algebraic Terms

Expand:

3x(x + 2)

Multiply:

3x(x) = 3x²

and:

3x(2) = 6x

Therefore:

3x(x + 2) = 3x² + 6x

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Powers During Multiplication

Remember:

x × x = x²

and:

x² × x = x³

For example:

2x²(3x + 4)

Distribute:

2x²(3x) + 2x²(4)

= 6x³ + 8x²

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More Than Two Terms

Distribution works with any number of terms inside parentheses.

For example:

3(x + 2y − 4)

Multiply 3 by every term:

3x + 6y − 12

Therefore:

3(x + 2y − 4) = 3x + 6y − 12

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Worked Example

Expand:

5(2x + 3y − 4)

Multiply 5 by each term:

10x + 15y − 20

Therefore:

5(2x + 3y − 4) = 10x + 15y − 20

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Distribution Followed by Combining Like Terms

Often distribution is only the first step.

Simplify:

3(x + 4) + 2x

Distribute:

3x + 12 + 2x

Combine like terms:

5x + 12

Therefore:

3(x + 4) + 2x = 5x + 12

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Worked Example

Simplify:

4(x + 2) + 3x + 5

Distribute:

4x + 8 + 3x + 5

Combine variable terms:

4x + 3x = 7x

Combine constants:

8 + 5 = 13

Therefore:

7x + 13

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Subtraction and Distribution

Simplify:

6x − 2(x + 5)

The factor being distributed is:

−2

So:

6x − 2x − 10

Combine:

4x − 10

Therefore:

6x − 2(x + 5) = 4x − 10

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Worked Example with Negative Distribution

Simplify:

5x − 3(x − 4)

Distribute −3:

5x − 3x + 12

Combine:

2x + 12

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Distribution on Both Sides

Consider:

2(x + 3) + 4(x − 1)

Distribute:

2x + 6 + 4x − 4

Combine:

6x + 2

Therefore:

2(x + 3) + 4(x − 1) = 6x + 2

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Equivalent Expressions

The distributive property creates equivalent expressions.

For example:

3(x + 5)

and:

3x + 15

look different, but they have the same value for every x.

If x = 4:

First expression:

3(4 + 5) = 27

Second expression:

3(4) + 15 = 27

Therefore they are equivalent.

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Checking Distribution by Substitution

Suppose we claim:

4(x − 3) = 4x − 12

Test x = 5.

Left:

4(5 − 3) = 8

Right:

4(5) − 12 = 8

The values agree.

Correct algebra proves the expressions are equivalent for all permissible x; substitution is useful for checking our work.

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Distribution and Order of Operations

Consider:

2(3 + 5)

Using parentheses first:

2(8) = 16

Using distribution:

2(3) + 2(5)

6 + 10 = 16

Both methods give the same result.

This demonstrates why the distributive property is consistent with arithmetic.

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When Is Distribution Useful?

Distribution is especially useful when:

• removing parentheses
• simplifying expressions
• solving equations
• multiplying algebraic terms
• working with area models
• factoring expressions
• evaluating expressions efficiently
• rearranging scientific formulas

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Distribution in Equations

Solve:

3(x + 4) = 24

Distribute:

3x + 12 = 24

Subtract 12:

3x = 12

Divide by 3:

x = 4

Check:

3(4 + 4) = 24

3(8) = 24

Correct.

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Distribution Before Solving

Solve:

2(x − 5) + 6 = 18

Distribute:

2x − 10 + 6 = 18

Combine:

2x − 4 = 18

Add 4:

2x = 22

Divide by 2:

x = 11

Check:

2(11 − 5) + 6 = 18

12 + 6 = 18

Correct.

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Variables on Both Sides

Solve:

3(x + 2) = 2x + 10

Distribute:

3x + 6 = 2x + 10

Subtract 2x:

x + 6 = 10

Subtract 6:

x = 4

Distribution allows us to remove the parentheses before collecting variable terms.

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Parentheses on Both Sides

Solve:

2(x + 5) = 4(x − 1)

Distribute:

2x + 10 = 4x − 4

Subtract 2x:

10 = 2x − 4

Add 4:

14 = 2x

Divide by 2:

x = 7

Check:

LHS:

2(7 + 5) = 24

RHS:

4(7 − 1) = 24

Correct.

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Distribution Can Sometimes Be Avoided

Distribution is useful, but it is not always the fastest method.

Consider:

5(x + 3) = 40

We could distribute:

5x + 15 = 40

and solve.

But we could also divide both sides by 5 first:

x + 3 = 8

Then:

x = 5

Both approaches are correct.

Efficient algebra involves choosing useful operations.

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Another Efficient Example

Solve:

4(2x − 1) = 28

Instead of immediately distributing, divide both sides by 4:

2x − 1 = 7

Add 1:

2x = 8

Divide by 2:

x = 4

Sometimes removing a common multiplier first makes an equation simpler.

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Factoring: Reversing Distribution

The distributive property also works in reverse.

Expansion:

3(x + 4) = 3x + 12

Factoring reverses this:

3x + 12 = 3(x + 4)

To factor, identify a factor shared by all terms.

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Factoring a Common Number

Factor:

4x + 12

Both terms are divisible by 4.

Factor out 4:

4(x + 3)

Check by distributing:

4(x + 3) = 4x + 12

Correct.

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Greatest Common Factor

The greatest common factor (GCF) is the largest factor shared by all terms.

Consider:

6x + 18

The greatest common factor is:

6

So:

6x + 18 = 6(x + 3)

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Worked Factoring Example

Factor:

8x + 20

The GCF of 8 and 20 is:

4

Therefore:

8x + 20 = 4(2x + 5)

Check:

4(2x + 5)

= 8x + 20

Correct.

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Factoring a Variable

Factor:

x² + 5x

Both terms contain x.

Factor out x:

x(x + 5)

Distributing confirms:

x(x + 5) = x² + 5x

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Factoring a Number and Variable

Factor:

6x² + 9x

Both terms contain:

3x

Factor it out:

3x(2x + 3)

Check:

3x(2x) + 3x(3)

= 6x² + 9x

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Factoring with Negative Terms

Factor:

−4x − 12

We can factor out −4:

−4(x + 3)

Check:

−4(x + 3)

= −4x − 12

Factoring out a negative can sometimes make the expression inside parentheses easier to work with.

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Expanding and Factoring Are Opposites

Expansion:

5(x + 2) → 5x + 10

Factoring:

5x + 10 → 5(x + 2)

These processes are closely related.

You can often check one by performing the other.

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Distribution in Mental Mathematics

The distributive property is useful even without variables.

Calculate:

7 × 23

Rewrite:

7(20 + 3)

Distribute:

7(20) + 7(3)

= 140 + 21

= 161

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Another Mental Math Example

Calculate:

6 × 98

Write:

6(100 − 2)

Distribute:

600 − 12

= 588

The same property used in algebra also helps with arithmetic.

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Distribution in Geometry

Suppose a rectangle has:

length = x + 5

and:

width = 3

Area:

A = 3(x + 5)

Distribute:

A = 3x + 15

Both expressions represent the same area.

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Distribution in Perimeter Problems

Suppose a rectangle has:

length = x + 4

width = x + 1

Perimeter:

P = 2(x + 4) + 2(x + 1)

Distribute:

P = 2x + 8 + 2x + 2

Combine:

P = 4x + 10

The simplified expression represents the rectangle's perimeter.

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Distribution in Cost Problems

Suppose three students each buy:

• one ticket costing x dollars
• one snack costing $4

The cost for one student is:

x + 4

For three students:

3(x + 4)

Distribute:

3x + 12

This represents:

• 3 tickets
• $12 total for snacks

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Distribution in Science

Suppose a quantity is described by:

E = m(cΔT + L)

The mass m multiplies both terms.

Distribute:

E = mcΔT + mL

Both forms are mathematically equivalent.

The most useful form depends on what we are trying to calculate or interpret.

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A Longer Simplification

Simplify:

4(2x + 3) − 2(x − 5)

Distribute:

8x + 12 − 2x + 10

Combine like terms:

6x + 22

Therefore:

4(2x + 3) − 2(x − 5) = 6x + 22

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A More Challenging Example

Simplify:

−3(2x − 4) + 5(x + 2)

Distribute:

−6x + 12 + 5x + 10

Combine:

−x + 22

Therefore:

−3(2x − 4) + 5(x + 2) = −x + 22

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Solving a More Complex Equation

Solve:

3(2x + 4) − 5 = 4x + 13

Distribute:

6x + 12 − 5 = 4x + 13

Combine:

6x + 7 = 4x + 13

Subtract 4x:

2x + 7 = 13

Subtract 7:

2x = 6

Divide by 2:

x = 3

Check using the original equation:

LHS:

3(2(3) + 4) − 5

= 3(10) − 5

= 25

RHS:

4(3) + 13

= 25

Correct.

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Recognizing Equivalent Expressions

Which expression is equivalent to:

5(2x − 3)?

Distribute:

10x − 15

Therefore:

5(2x − 3)

and:

10x − 15

are equivalent.

This type of reasoning is useful when comparing different algebraic forms.

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Checking by Factoring

Suppose we have:

12x + 18

Factor the GCF:

6(2x + 3)

Now distribute:

6(2x + 3)

= 12x + 18

This confirms the factorization.

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Common Mistakes

Mistake: Multiplying only the first term.

Incorrect:

3(x + 4) = 3x + 4

Correct:

3(x + 4) = 3x + 12

The outside factor multiplies every term inside.

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Mistake: Forgetting negative signs.

Incorrect:

−2(x + 5) = −2x + 10

Correct:

−2(x + 5) = −2x − 10

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Mistake: Mishandling two negatives.

Incorrect:

−3(x − 4) = −3x − 12

Correct:

−3(x − 4) = −3x + 12

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Mistake: Combining unlike terms after distribution.

3x + 12

cannot become:

15x

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Mistake: Changing exponents during addition.

2x + 5x = 7x

not:

7x²

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Mistake: Forgetting that −(x + 3) means −1(x + 3).

Correct:

−(x + 3) = −x − 3

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Mistake: Thinking distribution is always required.

Sometimes dividing out a common factor before distributing makes an equation easier.

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Mistake: Confusing factoring with combining like terms.

6x + 12

cannot be combined into 18x.

It can instead be factored:

6(x + 2)

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Key Terms

Distributive property — property stating that multiplication can be distributed across addition or subtraction: a(b + c) = ab + ac.

Distribute — multiply an outside factor by every term inside parentheses.

Expand — rewrite an expression without parentheses by carrying out multiplication.

Factor — number or algebraic expression multiplied by another quantity.

Factoring — rewriting an expression as a product of factors.

Greatest common factor (GCF) — largest factor shared by all terms.

Equivalent expressions — expressions that have the same value for every permissible value of their variables.

Term — number, variable, or product of numbers and variables within an expression.

Coefficient — numerical factor multiplying a variable.

Like terms — terms with identical variable parts.

Expression — combination of numbers, variables, and operations.

Area model — visual representation that uses rectangle areas to demonstrate multiplication and distribution.

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Key Takeaways

• The distributive property connects multiplication with addition and subtraction.
• The basic rule is a(b + c) = ab + ac.
• For subtraction, a(b − c) = ab − ac.
• The factor outside parentheses must multiply every term inside.
• Distribution removes parentheses and expands expressions.
• Positive and negative signs must be handled carefully.
• A negative outside parentheses affects every term inside.
• The expression −(x + 4) means −1(x + 4).
• Variables and algebraic terms can also be distributed.
• Distribution may create like terms that can then be combined.
• Distribution is often useful before solving equations.
• Distribution can help simplify equations containing variables on both sides.
• Sometimes dividing out a common factor before distributing is more efficient.
• Factoring is the reverse process of distribution.
• A common numerical factor can be removed from all terms.
• Variables can also be common factors.
• The GCF is often the most useful factor to remove.
• Expanding and factoring produce different-looking but equivalent expressions.
• The distributive property can be represented using rectangle area models.
• Distribution is useful in geometry, mental arithmetic, science, and real-world cost problems.
• Correct distribution is one of the foundations for later work with polynomials, factoring, and quadratic equations.

Check Your Understanding

1. State the distributive property using variables.
2. Explain in your own words what it means to distribute.
3. Expand: 3(x + 5).
4. Expand: 6(x − 4).
5. Expand: −2(x + 7).
6. Expand: −4(x − 3).
7. Expand: 5(2x + 3).
8. Expand: 3(x + 2y − 4).
9. Simplify: 2(x + 4) + 3x.
10. Simplify: 5(x − 2) + 4x + 3.
11. Simplify: 7x − 2(x + 5).
12. Simplify: −3(x − 4) + 5x.
13. Simplify: 4(2x + 3) − 2(x − 5).
14. Factor completely: 6x + 18.
15. Factor completely: 8x + 20.
16. Factor: x² + 7x.
17. Factor: 6x² + 9x.
18. Solve: 3(x + 5) = 27.
19. A rectangle has width 4 and length x + 6. Write its area in factored form and expanded form.
20. Explain why expanding 4(x + 3) and factoring 4x + 12 are reverse processes.