Learning Objectives
  • Expand single, double and repeated brackets.
  • Collect like terms after expansion.
  • Factorise by taking common factors and grouping.
  • Use perfect-square and difference-of-squares identities.
  • Factorise quadratic expressions fully and check by re-expansion.
Key Terms
Expand
Remove brackets by multiplication.
Factorise
Rewrite an expression as a product of factors.
Common factor
A factor contained in every term.
Quadratic expression
An expression whose highest variable power is 2.
Perfect square
An expression of the form (a\pm b)^2.
Difference of squares
An expression of the form a^2-b^2.
8th Edition Chapter Map
  • Distributive law and collecting like terms
  • Products of two or more brackets
  • Common-factor and grouping methods
  • Algebraic identities
  • Quadratic factorisation
The Distributive Law

Multiply every term inside a bracket by the factor outside. Signs must be handled carefully.

a(b+c)=ab+ac
Worked Example: Single Bracket

Question: Expand and simplify 4x(3x-5)+2x(x+7).

  1. 4x(3x-5)=12x^2-20x.
  2. 2x(x+7)=2x^2+14x.
  3. Collect like terms: 14x^2-6x.

Answer: 14x^2-6x.

Two Binomials

Every term in the first bracket multiplies every term in the second. A useful check is that the product of the first terms gives the leading term and the product of the constants gives the constant term.

Worked Example: Double Brackets

Question: Expand (3x-4)(2x+5).

  1. First terms: 6x^2.
  2. Cross terms: 15x-8x=7x.
  3. Constants: -20.

Answer: 6x^2+7x-20.

More Than Two Brackets

The syllabus can include products of more than two brackets. Expand two brackets first, simplify, then multiply by the remaining bracket.

Worked Example: Three Brackets

Question: Expand (x-2)(x+3)(2x+1).

  1. First, (x-2)(x+3)=x^2+x-6.
  2. Multiply by (2x+1).
  3. (x^2+x-6)(2x+1)=2x^3+3x^2-11x-6.

Answer: 2x^3+3x^2-11x-6.

Factorising By A Common Factor

Find the greatest numerical and algebraic factor common to every term. Factorise fully, not partially.

Worked Example: Greatest Common Factor

Question: Factorise 18x^3y-24x^2y^2.

  1. The numerical HCF is 6.
  2. The common variable factor is x^2y.
  3. Divide each term by 6x^2y.

Answer: 6x^2y(3x-4y).

Factorising By Grouping

Group terms so that each pair has a common factor and the remaining bracket is the same.

Worked Example: Four Terms

Question: Factorise ax+ay+bx+by.

  1. Group: a(x+y)+b(x+y).
  2. Take the common bracket (x+y).

Answer: (a+b)(x+y).

Important Identities
(a+b)^2=a^2+2ab+b^2
(a-b)^2=a^2-2ab+b^2
a^2-b^2=(a-b)(a+b)
Worked Example: Difference Of Squares

Question: Factorise 49x^2-81y^2.

  1. 49x^2=(7x)^2 and 81y^2=(9y)^2.
  2. Use a^2-b^2=(a-b)(a+b).

Answer: (7x-9y)(7x+9y).

Monic Quadratics

For x^2+bx+c, find two numbers whose sum is b and product is c.

Worked Example: Monic Quadratic

Question: Factorise x^2-7x+12.

  1. Find two numbers with product 12 and sum -7.
  2. They are -3 and -4.

Answer: (x-3)(x-4).

Non-Monic Quadratics

For ax^2+bx+c, use splitting the middle term. Find two numbers with product ac and sum b, then factorise by grouping.

Worked Example: Split The Middle Term

Question: Factorise 6x^2+x-2.

  1. ac=6(-2)=-12. Numbers with product -12 and sum 1 are 4 and -3.
  2. Rewrite: 6x^2+4x-3x-2.
  3. Group: 2x(3x+2)-1(3x+2).

Answer: (2x-1)(3x+2).

Expressions With A Common Variable Factor

Expressions such as ax^3+bx^2+cx should first be factorised by extracting x. The remaining quadratic may factorise further.

Worked Example: Factorise Fully

Question: Factorise 2x^3-5x^2-3x.

  1. Extract x: x(2x^2-5x-3).
  2. Factorise the quadratic: 2x^2-5x-3=(2x+1)(x-3).

Answer: x(2x+1)(x-3).

Examination Guidance
  • After factorising, multiply the brackets back out as a check.
  • Factorise fully: always look for a common factor before any other method.
  • Keep signs attached to terms during expansion.
  • Use identities when the structure matches; this is faster than general multiplication.
Common Mistakes
  • Writing (a-b)^2=a^2-b^2; the middle term is missing.
  • Failing to multiply every term when expanding.
  • Stopping after taking out only part of the common factor.
  • Choosing numbers with the correct product but the wrong sum in a quadratic.
Knowledge Check And Practice

1. Expand (2x-3)(x+4).

Answer: 2x^2+5x-12.

2. Factorise 15a^2b-20ab^2.

Answer: 5ab(3a-4b).

3. Factorise x^2+9x+20.

Answer: (x+4)(x+5).

4. Factorise 9p^2-25q^2.

Answer: (3p-5q)(3p+5q).

5. Factorise fully 3x^3+6x^2-24x.

Answer: 3x(x+4)(x-2).
Extended Worked Practice
Expanding Three Factors

Question: Expand (x+1)(x-2)(x+3).

  1. First, (x+1)(x-2)=x^2-x-2.
  2. Multiply by (x+3): (x^2-x-2)(x+3).
  3. =x^3+3x^2-x^2-3x-2x-6=x^3+2x^2-5x-6.

Answer: x^3+2x^2-5x-6.

Factorising By Grouping And Difference Of Squares

Question: Factorise 3x^3-12x completely.

  1. Take out the common factor 3x: 3x(x^2-4).
  2. Recognise a difference of squares: x^2-4=(x-2)(x+2).

Answer: 3x(x-2)(x+2).

Factorising A Non-Monic Quadratic

Question: Factorise 6x^2+11x+3.

  1. Find two numbers with product 6\times3=18 and sum 11: 9 and 2.
  2. Split the middle term: 6x^2+9x+2x+3.
  3. Group: 3x(2x+3)+1(2x+3).

Answer: (3x+1)(2x+3).

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