The Distributive Property

The distributive property is an important algebraic rule that allows us to remove brackets by multiplying the term outside the brackets by every term inside.

For example:

3(x + 4)

means that 3 multiplies both x and 4:

3(x + 4) = 3x + 12

This process is often called expanding brackets.

Learning Targets

• I can expand brackets using the distributive property.
• I can distribute negative signs correctly.
• I can simplify expressions after expansion.
• I can recognize when distribution is required.
• I can apply distribution in multi-step problems.

---

What Is the Distributive Property?

The distributive property tells us:

a(b + c) = ab + ac

The value outside the brackets multiplies every term inside the brackets.

For subtraction:

a(b − c) = ab − ac

For example:

5(x + 2)

Multiply 5 by x:

5 × x = 5x

Multiply 5 by 2:

5 × 2 = 10

Therefore:

5(x + 2) = 5x + 10

---

Why Does Distribution Work?

We can understand distribution using ordinary numbers.

Consider:

3(4 + 2)

Calculate inside the brackets first:

3(6) = 18

Now distribute instead:

3(4) + 3(2)

= 12 + 6

= 18

Both methods give the same result.

Therefore:

3(4 + 2) = 3(4) + 3(2)

The same principle works when variables are involved.

---

An Area Model

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

Imagine a rectangle with height:

3

and total width:

x + 4

Its area is:

3(x + 4)

We can divide the rectangle into two smaller rectangles.

One has area:

3x

The other has area:

3 × 4 = 12

Therefore, the total area is:

3x + 12

So:

3(x + 4) = 3x + 12

This provides a visual explanation of why distribution works.

---

Expanding Brackets

To expand brackets, multiply the term outside the brackets by every term inside.

Consider:

4(x + 5)

Multiply:

4 × x = 4x

and:

4 × 5 = 20

Therefore:

4(x + 5) = 4x + 20

Once the multiplication has been completed, the brackets are no longer needed.

---

Worked Example — Addition Inside Brackets

Expand:

6(x + 3)

Multiply 6 by x:

6x

Multiply 6 by 3:

18

Therefore:

6(x + 3) = 6x + 18

---

Worked Example — Subtraction Inside Brackets

Expand:

5(x − 4)

Multiply 5 by x:

5x

Multiply 5 by −4:

−20

Therefore:

5(x − 4) = 5x − 20

The subtraction sign stays with the term being multiplied.

---

Every Term Must Be Multiplied

A common mistake is multiplying only the first term.

For example:

3(x + 7)

Incorrect:

3x + 7

Correct:

3x + 21

because:

3 × x = 3x

and:

3 × 7 = 21

A useful visual technique is to draw arrows from the outside term to each term inside the brackets.

---

Distributing to Several Terms

The brackets may contain more than two terms.

For example:

2(x + y + 5)

Multiply 2 by every term:

2 × x = 2x

2 × y = 2y

2 × 5 = 10

Therefore:

2(x + y + 5) = 2x + 2y + 10

The distributive property works no matter how many terms are inside the brackets.

---

Variables Outside the Brackets

The term outside the brackets can contain a variable.

Consider:

x(x + 4)

Distribute x:

x × x = x²

x × 4 = 4x

Therefore:

x(x + 4) = x² + 4x

Another example:

3x(x + 2)

Multiply:

3x × x = 3x²

3x × 2 = 6x

Therefore:

3x(x + 2) = 3x² + 6x

---

Distributing Negative Numbers

Extra care is required when the term outside the brackets is negative.

Consider:

−3(x + 4)

Multiply −3 by x:

−3x

Multiply −3 by 4:

−12

Therefore:

−3(x + 4) = −3x − 12

Remember the multiplication rules for signs:

Positive × Positive = Positive

Positive × Negative = Negative

Negative × Positive = Negative

Negative × Negative = Positive

---

Negative Times Negative

Consider:

−4(x − 3)

Multiply −4 by x:

−4x

Now multiply:

−4 × −3 = +12

Therefore:

−4(x − 3) = −4x + 12

The second sign changes because multiplying two negative numbers produces a positive number.

---

Worked Example — Negative Distribution

Expand:

−5(2x − 7)

Multiply −5 by 2x:

−10x

Multiply −5 by −7:

+35

Therefore:

−5(2x − 7) = −10x + 35

This is an important situation where careful attention to signs is essential.

---

A Negative Sign Before Brackets

Sometimes there is only a negative sign before the brackets.

For example:

−(x + 5)

A negative sign can be thought of as multiplication by:

−1

Therefore:

−1(x + 5)

Distribute −1:

−x − 5

So:

−(x + 5) = −x − 5

---

Subtracting an Entire Expression

Consider:

10 − (x + 3)

The negative sign applies to everything inside the brackets.

Rewrite:

10 − x − 3

Combine the constants:

7 − x

Therefore:

10 − (x + 3) = 7 − x

---

Subtracting an Expression with a Negative Term

Consider:

12 − (2x − 5)

Distribute the negative sign:

12 − 2x + 5

Notice that:

−(−5) = +5

Now combine constants:

17 − 2x

Therefore:

12 − (2x − 5) = 17 − 2x

---

Expand and Simplify

Often, expanding brackets is only the first step.

After expansion, we may need to combine like terms.

Consider:

3(x + 4) + 2x

First expand:

3x + 12 + 2x

Now combine like terms:

3x + 2x = 5x

Therefore:

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

---

Worked Example — Expand and Simplify

Simplify:

4(x + 2) + 3x

Expand:

4x + 8 + 3x

Combine like terms:

4x + 3x = 7x

Therefore:

4(x + 2) + 3x = 7x + 8

---

Worked Example — Including Subtraction

Simplify:

5(x − 3) + 2x

Expand:

5x − 15 + 2x

Combine like terms:

5x + 2x = 7x

Therefore:

5(x − 3) + 2x = 7x − 15

---

Worked Example — Negative Distribution and Simplifying

Simplify:

8x − 3(x + 2)

Distribute −3:

8x − 3x − 6

Combine like terms:

8x − 3x = 5x

Therefore:

8x − 3(x + 2) = 5x − 6

The negative sign is part of the number being distributed.

---

Two Sets of Brackets

Sometimes an expression contains more than one set of brackets.

Consider:

2(x + 3) + 4(x − 1)

Expand the first brackets:

2x + 6

Expand the second brackets:

4x − 4

Now write everything together:

2x + 6 + 4x − 4

Combine like terms:

2x + 4x = 6x

and:

6 − 4 = 2

Therefore:

2(x + 3) + 4(x − 1) = 6x + 2

---

Worked Example — Two Brackets with Subtraction

Simplify:

5(x + 2) − 2(x − 3)

Expand the first brackets:

5x + 10

Distribute −2 through the second brackets:

−2x + 6

Now:

5x + 10 − 2x + 6

Combine like terms:

5x − 2x = 3x

10 + 6 = 16

Therefore:

5(x + 2) − 2(x − 3) = 3x + 16

---

Distribution with Several Variables

The distributive property works with several variables.

Consider:

3x(2x + y − 4)

Multiply 3x by every term:

3x × 2x = 6x²

3x × y = 3xy

3x × −4 = −12x

Therefore:

3x(2x + y − 4) = 6x² + 3xy − 12x

The same rule applies:

Multiply every term inside the brackets.

---

Recognizing When Distribution Is Required

Distribution is needed when a number, variable, or term multiplies an entire group.

Examples include:

3(x + 5)

−2(x − 7)

x(x + 4)

5a(2a + 3)

You may also need distribution when subtracting an entire expression:

8 − (x + 2)

In each case, the multiplication applies to every term in the group.

---

When Distribution Is Not Required

Not every set of brackets needs to be expanded immediately.

For example:

3(x + 4)

may sometimes be more useful in its factored form.

If you only need to evaluate it when:

x = 2

you could calculate:

3(2 + 4)

= 3(6)

= 18

There is no requirement to expand first.

Whether expansion is useful depends on the problem.

---

Equivalent Expressions

Expansion creates an expression that looks different but has the same value.

For example:

4(x + 3)

and:

4x + 12

are equivalent expressions.

We can check this by choosing a value.

Let:

x = 5

First expression:

4(5 + 3)

= 4(8)

= 32

Second expression:

4(5) + 12

= 20 + 12

= 32

Both expressions give the same result.

---

Distribution and Factoring Are Opposites

Distribution expands an expression:

3(x + 4) → 3x + 12

Factoring reverses the process:

3x + 12 → 3(x + 4)

These are opposite ways of writing the same relationship.

This becomes especially useful later when working with equations, quadratics, and algebraic fractions.

---

Distribution in Equations

The distributive property is also important when solving equations.

Consider:

3(x + 2) = 18

One method is to expand:

3x + 6 = 18

Subtract 6:

3x = 12

Divide by 3:

x = 4

We can check:

3(4 + 2)

= 3(6)

= 18

So the solution is correct.

---

Worked Example — Multi-Step Equation

Solve:

2(x + 5) + 3x = 25

First expand:

2x + 10 + 3x = 25

Combine like terms:

5x + 10 = 25

Subtract 10:

5x = 15

Divide by 5:

x = 3

Check:

2(3 + 5) + 3(3)

= 2(8) + 9

= 16 + 9

= 25

Therefore:

x = 3

---

Distribution in Geometry

The distributive property appears naturally when calculating areas.

Suppose a rectangle has:

Height = 5 cm

and:

Width = x + 3 cm

The area is:

A = 5(x + 3)

Expand:

A = 5x + 15

Both expressions describe the same area:

5(x + 3)

and:

5x + 15

---

Distribution in Everyday Situations

Suppose three students each buy:

• One notebook costing x dollars
• One pen costing $2

The cost for one student is:

x + 2

The cost for three students is:

3(x + 2)

Using distribution:

3x + 6

This tells us that the total consists of:

3 notebooks costing 3x

and:

3 pens costing $6

Distribution can therefore help us interpret real-world situations.

---

A Useful Expansion Strategy

When expanding an expression:

Look at the term outside the brackets.

Then:

Multiply it by the first term.

Multiply it by the second term.

Continue until every term has been multiplied.

Then:

Remove the brackets.

Finally:

Combine like terms if possible.

For negative terms, check the signs carefully before moving on.

---

Checking an Expansion

Suppose you want to check:

3(x − 4) = 3x − 12

Choose:

x = 6

Left side:

3(6 − 4)

= 3(2)

= 6

Right side:

3(6) − 12

= 18 − 12

= 6

Both sides give the same result.

Substitution can therefore provide a useful check.

---

Common Mistakes

Mistake: Multiplying only the first term

Incorrect:

4(x + 3) = 4x + 3

Correct:

4(x + 3) = 4x + 12

The 4 must multiply every term inside the brackets.

Mistake: Forgetting the negative sign

Incorrect:

−3(x + 2) = −3x + 6

Correct:

−3(x + 2) = −3x − 6

Mistake: Mishandling two negatives

Incorrect:

−2(x − 5) = −2x − 10

Correct:

−2(x − 5) = −2x + 10

because:

Negative × Negative = Positive

Mistake: Forgetting that a lone negative sign means −1

−(x + 4)

means:

−1(x + 4)

Therefore:

−x − 4

Mistake: Stopping after expansion

Consider:

3(x + 2) + 4x

Expanding gives:

3x + 6 + 4x

but this can still be simplified:

7x + 6

Mistake: Combining unlike terms

3x + 6

cannot be simplified to:

9x

The terms are unlike.

---

Key Terms

Distributive property – the rule that multiplication over a grouped sum or difference can be applied to every term in the group.

Expand – remove brackets by applying multiplication to every term inside.

Bracket – a grouping symbol used to show that terms belong together.

Term – one part of an algebraic expression.

Coefficient – the numerical factor multiplying a variable.

Like terms – terms with identical variable parts.

Simplify – rewrite an expression in a simpler equivalent form.

Equivalent expressions – expressions that have the same value for every allowed value of their variables.

Factor – a quantity multiplied by another quantity.

Factoring – rewriting an expression as a product; the reverse of expansion.

---

Check Your Understanding

1. Expand:

3(x + 5)

2. Expand:

6(x − 2)

3. Expand:

−4(x + 3)

4. Expand:

−5(x − 2)

5. Expand:

2(3x + 4)

6. Expand:

x(x + 6)

7. Simplify:

3(x + 4) + 2x

8. Simplify:

5(x − 2) + 3x

9. Simplify:

7x − 2(x + 5)

10. Simplify:

4(x + 3) + 2(x − 1)

11. Simplify:

5(x + 2) − 3(x − 4)

12. Simplify:

10 − (2x − 3)

13. Explain the error:

4(x + 5) = 4x + 5

14. Explain why:

−3(x − 4) = −3x + 12

15. Solve:

3(x + 2) = 21

16. Solve:

2(x + 4) + 3x = 23

---

Key Takeaways

• The distributive property allows us to expand brackets.
• The term outside the brackets must multiply every term inside.
• a(b + c) = ab + ac
• a(b − c) = ab − ac
• A negative value must also be distributed to every term.
• Multiplying two negative values produces a positive value.
• A negative sign immediately before brackets can be treated as multiplication by −1.
• After expanding, look for like terms that can be combined.
• Expressions can contain several sets of brackets that must each be expanded.
• Distribution is useful when simplifying expressions and solving equations.
• Expansion produces an expression that is equivalent to the original expression.
• Substitution can be used to check whether an expansion is correct.
• Distribution and factoring are reverse processes.
• The distributive property appears in algebra, geometry, formulas, and real-world calculations.

Did You Know?

The distributive property is built into ordinary mental arithmetic. For example, you might calculate:

7 × 19

by thinking:

7(20 − 1)

Then distribute:

140 − 7 = 133

So even when people do not call it the distributive property, they often use it naturally to make calculations easier.