Expanding, factorising & simplifying expressions

Algebraic expressions can be rewritten in equivalent forms. Expanding means removing brackets by multiplying out — using the distributive property, a(b + c) = ab + ac. Factorising is the reverse: pulling a common factor back out and writing the expression with brackets, ab + ac = a(b + c). Simplifying means combining like terms (terms with the same variable part) to write an expression as concisely as possible. These moves rely on number properties you can trust: the distributive property (splitting multiplication over addition), the commutative property (order doesn't matter for + or ×), and the associative property (grouping doesn't matter for + or ×).

Key terms

Expand:

To remove brackets by multiplying out using the distributive property.

Factorise:

To rewrite an expression using a common factor pulled out into brackets.

Like terms:

Terms that share the same variable part and can be combined.

Questions

1. 1. Expanding an expression means:

   • Removing brackets by multiplying out
   • Adding brackets in
   • Deleting all variables
   • Rounding the answer

2. 2. Factorising is the reverse of:

   • Expanding
   • Simplifying only
   • Solving equations
   • Graphing

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

   • 2x + 6
   • 2x + 3
   • x + 6
   • 2x + 5

4. 4. Like terms share the same:

   • Variable part
   • Sign only
   • Number of digits
   • Colour

5. 5. Simplify 3x + 2x:

   • 5x
   • 6x
   • 5x²
   • 5

6. 6. Factorise 4x + 8:

   • 4(x + 2)
   • 4(x + 8)
   • 2(x + 4)
   • x(4 + 8)

7. 7. The distributive property lets you:

   • Split multiplication over addition, e.g. a(b+c) = ab+ac
   • Ignore brackets completely
   • Always divide instead of multiply
   • Swap addition for subtraction freely

8. 8. Expand 5(2x + 3):

   • 10x + 15
   • 10x + 3
   • 7x + 15
   • 2x + 15

9. 9. Simplify 7x + 4 − 2x + 1:

   • 5x + 5
   • 9x + 5
   • 5x + 3
   • 9x + 3

10. 10. Factorise 6x + 9:

    • 3(2x + 3)
    • 3(2x + 9)
    • 6(x + 9)
    • x(6 + 9)

11. 11. Expand −2(x − 4):

    • −2x + 8
    • −2x − 8
    • 2x + 8
    • −2x + 4

12. 12. Simplify 4x² + 3x − x² + 2x:

    • 3x² + 5x
    • 3x² + x
    • 4x² + 5x
    • 3x + 5x²

13. 13. Factorise 10x − 15:

    • 5(2x − 3)
    • 5(2x − 15)
    • 10(x − 15)
    • x(10 − 15)

14. 14. The commutative property means, for addition and multiplication:

    • Order does not matter
    • Order always matters
    • Only subtraction is commutative
    • Brackets must always be used

15. 15. Expand and simplify 3(x + 2) + 2(x + 1):

    • 5x + 8
    • 5x + 3
    • 3x + 8
    • 5x + 6

16. 16. Factorise fully: 12x² + 18x:

    • 6x(2x + 3)
    • 6(2x² + 3x)
    • 2x(6x + 9)
    • 3x(4x + 6)

17. 17. Why is factorising sometimes described as the "reverse" of expanding?

    • Expanding removes brackets by multiplying out, while factorising rebuilds brackets from a common factor
    • They are actually the exact same operation
    • Factorising always makes an expression longer than expanding
    • There is no mathematical relationship between the two

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

    • 2x − 10
    • 2x − 4
    • 6x − 10
    • 2x + 10

19. 19. Why might factorising an expression make it easier to solve an equation like x² + 5x = 0?

    • Factorising to x(x + 5) = 0 lets you use the fact that if a product is 0, one of the factors must be 0
    • Factorising never helps with solving equations
    • Factorising always makes equations impossible to solve
    • You must always expand before solving any equation

20. 20. Simplify 2(3x + 1) − (x − 4):

    • 5x + 6
    • 5x − 2
    • 7x + 6
    • 5x + 2

21. 21. Understanding how to expand, factorise and simplify mainly helps you to:

    • Rewrite algebraic expressions in equivalent but more useful forms
    • Always keep expressions in their original form with no changes
    • Avoid ever using brackets in algebra
    • Treat every algebraic expression as unsolvable