Algebra

Expanding and Factorising with Single Brackets

Year 7 · Year 8 · Year 9

✓By the end of this lesson students will be able to expand expressions involving a single bracket.
✓By the end of this lesson students will be able to multiply a single term by an expression in a bracket.
✓By the end of this lesson students will be able to factorise expressions by taking out a common factor.
✓By the end of this lesson students will be able to identify the Highest Common Factor (HCF) of algebraic terms.

Key concepts

Expanding a Single Bracket

Expanding a single bracket means multiplying the term outside the bracket by each term inside the bracket. This is based on the distributive law. For example, a(b + c) means 'a multiplied by b' PLUS 'a multiplied by c'.

a(b + c) = ab + ac

Factorising into a Single Bracket

Factorising is the reverse process of expanding. It involves finding the Highest Common Factor (HCF) of all the terms in an expression and writing the expression as a product of the HCF and another expression in a bracket. Essentially, you are 'taking out' the common factor.

Key facts to remember

1Expanding means removing the brackets by multiplying.
2Factorising means putting an expression into brackets by finding a common factor.
3The distributive law is key to expanding: a(b + c) = ab + ac.
4When factorising, always look for the Highest Common Factor (HCF) of all terms.
5A common factor can be a number, a variable, or both.
6Multiplying two negative numbers results in a positive number.
7Multiplying a positive and a negative number results in a negative number.

Worked examples

Example 1

Expand 4(x + 7).

IMultiply the term outside the bracket (4) by the first term inside the bracket (x): 4 × x = 4x.

IIMultiply the term outside the bracket (4) by the second term inside the bracket (7): 4 × 7 = 28.

IIICombine the results with the correct sign.

Answer

4x + 28

Remember to multiply the term outside by EVERY term inside the bracket.

Example 2

Expand -3(2y - 5).

IMultiply -3 by the first term inside the bracket (2y): -3 × 2y = -6y.

IIMultiply -3 by the second term inside the bracket (-5): -3 × -5 = +15.

IIICombine the results.

Answer

-6y + 15

Be very careful with negative signs. A negative multiplied by a negative gives a positive.

Example 3

Factorise 12x + 18.

IFind the Highest Common Factor (HCF) of 12x and 18. The factors of 12 are 1, 2, 3, 4, 6, 12. The factors of 18 are 1, 2, 3, 6, 9, 18. The HCF is 6.

IIWrite the HCF outside a bracket: 6(...).

IIIDivide each term in the original expression by the HCF: 12x ÷ 6 = 2x and 18 ÷ 6 = 3.

IVPlace these results inside the bracket.

Answer

6(2x + 3)

You can check your answer by expanding 6(2x + 3) to see if you get 12x + 18.

Common mistakes

✗Forgetting to multiply ALL terms inside the bracket when expanding (e.g., 3(x + 2) = 3x + 2 instead of 3x + 6).
✗Incorrectly handling negative signs when expanding (e.g., -2(x - 3) = -2x - 6 instead of -2x + 6).
✗Not finding the HIGHEST Common Factor when factorising (e.g., factorising 4x + 8 as 2(2x + 4) instead of 4(x + 2)).
✗Leaving a common factor inside the bracket after factorising (e.g., 5x + 10 = 5(x + 10/5) which is 5(x+2), but sometimes students make errors here).
✗Confusing expanding and factorising; they are inverse operations.

Exam tips

★Always show your working step-by-step, especially with negative numbers.
★Double-check your signs carefully when expanding or factorising.
★To check your factorised answer, expand it to see if you get the original expression.
★Practise regularly with a variety of questions to build confidence and speed.

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