Practice Maths

Expanding with the Distributive Law

Key Ideas

distributive law
The rule a(b + c) = ab + ac — every term inside the brackets is multiplied by the factor outside.
expand
Remove brackets by applying the distributive law, multiplying through by the outside factor.
coefficient
The numerical factor in a term — in 5x, the coefficient is 5; in −3x, it is −3.
Hot Tip A common mistake is to only multiply the first term in the bracket. Remember: the factor outside multiplies every term inside the brackets.

Worked Example

Expand and simplify: 3(2x + 5) − 2(4x − 1)

Step 1 — Expand each bracket.
3(2x + 5) = 6x + 15
−2(4x − 1) = −8x + 2

Step 2 — Collect like terms.
6x − 8x + 15 + 2 = −2x + 17

Why the Distributive Law Works: The Area Model

The distributive law a(b + c) = ab + ac has a beautiful geometric explanation. Imagine a rectangle with width a and length (b + c). Its total area is a(b + c). But split the rectangle into two parts: one with dimensions a × b and one with dimensions a × c. The total area is also ab + ac. Both must be equal because they're the same rectangle — so a(b + c) = ab + ac. This geometric proof shows the law isn't just a rule to memorise; it reflects a fundamental truth about area.

In practice, expanding means multiplying the term outside the brackets by every term inside. For 4(3x − 2): multiply 4 by 3x to get 12x, and multiply 4 by −2 to get −8. Result: 12x − 8. Never stop after multiplying just the first term.

Expanding with Negative Signs Outside

The trickiest case is a negative factor outside. Consider −3(x − 4). Multiply −3 by x: −3x. Multiply −3 by −4: +12. Result: −3x + 12. The double negative (−3 × −4) produces a positive. This is where most mistakes happen — students forget that the negative distributes to ALL terms including the negative ones inside the bracket.

Another way to think about it: −3(x − 4) = −3(x + (−4)) = −3 × x + (−3) × (−4) = −3x + 12. Writing out the negatives explicitly can help prevent errors.

Key Tip: When a negative is outside the bracket, it changes the sign of every term inside. −5(2x − 3) = −10x + 15. The − sign "flips" every sign inside. A common error is getting −10x − 15 by forgetting to flip the second sign.

Expanding Two Sets of Brackets and Collecting Like Terms

When two expanded expressions are combined: 3(2x + 5) − 2(4x − 1). Step 1, expand each: 6x + 15 and −8x + 2. Step 2, write together: 6x + 15 − 8x + 2. Step 3, collect like terms: (6x − 8x) + (15 + 2) = −2x + 17. The minus sign before the second bracket means we're subtracting everything inside — equivalently, multiplying the second bracket by −2, not +2.

A useful check: substitute a simple number (like x = 1) into both the original and expanded expressions to see if they give the same answer. If 3(2(1) + 5) − 2(4(1) − 1) = 3(7) − 2(3) = 21 − 6 = 15, then −2(1) + 17 = −2 + 17 = 15. Match! The expansion is correct.

Key Tip: After expanding and simplifying, verify your answer by substituting a simple value (x = 0 or x = 1) into both the original and your result. If both give the same number, your expansion is correct. This is a quick and reliable self-check.

Expanding with Variable Factors

The factor outside can itself be a variable or a product: x(x + 3) = x2 + 3x. Here x × x = x2 (not 2x!). And 2x(3x − 5) = 6x2 − 10x. When multiplying: coefficients multiply, and variable powers add (using the multiplication index law). This is the first step toward more advanced algebra in Years 9 and 10.

Mastery Practice

  1. Expand using the distributive law. Fluency

    1. 3(x + 4)
    2. 5(2a + 3)
    3. 4(m − 7)
    4. 2(3b + 8)
    5. 6(n + 5)
    6. 7(2x − 3)
    7. 9(p + 1)
    8. 4(5k − 2)

  2. Expand. Take care with the negative sign. Fluency

    1. −3(x + 4)
    2. −2(5a − 1)
    3. −4(m + 6)
    4. −(n − 8)
    5. −5(2b + 3)
    6. −7(x − 2)
    7. −3(4y + 5)
    8. −6(3p − 4)

  3. Expand and then collect like terms to simplify fully. Fluency

    1. 2(x + 3) + 4x
    2. 3(2a + 1) − 5a
    3. 4(m + 2) + 3(m + 1)
    4. 5(b + 4) − 2(b − 3)
    5. 3(n + 5) + 2(4n − 1)
    6. 6(x − 2) − 3(2x + 1)
    7. 4(3y + 2) + 5(2y − 3)
    8. 2(5k − 4) − 3(k + 6)

  4. Expand each expression, then verify your answer is correct by substituting x = 2. Both the original and expanded form should give the same value. Understanding

    1. 4(x + 3)
    2. 2(3x − 5)
    3. −3(x + 4)
    4. 5(2x − 1) + 3x
    5. 2(x + 6) − 4(x − 1)
    6. 3(2x + 1) − 2(3x − 4)

  5. Find the missing value (represented by ?) in each equation. Understanding

    1. ?(x + 3) = 4x + 12
    2. 5(x + ?) = 5x + 20
    3. ?(x − 2) = 3x − 6
    4. ?(x + 4) = −2x − 8
    5. 3(? + 5) = 3x + 15 (find the missing pronumeral)
    6. ?(2x + 1) = 6x + 3
    7. 4(3x − ?) = 12x − 20
    8. −?(x + 6) = −5x − 30

  6. Apply the distributive law to real-world contexts. Problem Solving

    1. A rectangular room has a length of (x + 5) m and width of 4 m. Write and expand an expression for its area. Find the area when x = 3.
    2. A plumber charges $80 per hour plus $50 for materials. She works for (n + 2) hours. Write and expand an expression for the total cost. Find the cost when n = 3.
    3. A garden has two rectangular sections. Section A is 3 m by (x + 2) m and Section B is 5 m by (x − 1) m. Write an expression for the total area and simplify. Evaluate when x = 6.
    4. Explain why 2(x + 3) is not equal to 2x + 3. Use substitution to show your reasoning.

  7. Expand each expression where the term outside the brackets includes a variable. Fluency

    1. x(x + 3)
    2. 2a(3a − 4)
    3. m(5m + 2)
    4. 3y(2y − 7)
    5. 4b(b + 6)
    6. −n(3n − 5)
    7. 5k(2k + 1)
    8. −2t(4t − 3)

  8. Expand each expression by distributing over three terms. Fluency

    1. 2(x + 3y − 4)
    2. 5(a − 2b + 7)
    3. −3(m + 4n − 1)
    4. 4(2p + q − 6)
    5. −(x − 3y + 2)
    6. 6(3a − b + 5)

  9. Expand all brackets, then collect like terms to simplify fully. Understanding

    1. 3(x + 2) + 4(x − 1) − 2x
    2. 2(3a − 4) − (5a + 2) + 3a
    3. 5(n + 1) − 3(n − 2) + 4(−n + 3)
    4. −2(x + 4) + 3(2x − 1) − 4(x + 5)
    5. 4(a + b) − 3(a − 2b) + a
    6. 2x(x + 3) − x(x − 1) + 4x

  10. Apply the distributive law to solve multi-step problems. Problem Solving

    1. A school buys n packs of coloured pencils. Each pack contains 12 red, 8 blue and 5 green pencils. Write and expand an expression for the total number of pencils. If n = 6, find the total.
    2. A square pool has side length (x + 4) m. Write and expand an expression for its perimeter. Find the perimeter when x = 3.
    3. The cost of a school camp is $200 per student plus a $50 bus fee per group of students. If there are (n + 5) students, write and expand the total cost expression. Evaluate when n = 15.
    4. Two students expand 3(2x − 4):
      • Student A gets: 6x − 4
      • Student B gets: 6x − 12
      Who is correct? Explain the error made by the incorrect student.
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