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Factorising

FoundationHigherAQAEdexcelOCR

Master factorising for GCSE Maths with structured, exam-style practice. This Foundation and Higher resource covers factorising expressions and includes a complete mark scheme showing the steps examiners reward. Suitable for AQA, Edexcel and OCR. Free to download as a PDF. Take out the highest common factor and place it outside the bracket.

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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.

Foundation

WorksheetFoundation · PDFMark SchemeFoundation · PDF

Higher

WorksheetHigher · PDFMark SchemeHigher · PDF

Topic overview

Factorising is the reverse of expanding. Instead of removing brackets you put them back in, by taking out a common factor from every term.

The factor you take out must divide into all the terms. In \(6x + 9\), both terms divide by \(3\), so it factorises to \(3(2x + 3)\). Letters can be common factors too, so \(x^2 + 5x\) becomes \(x(x + 5)\).

Always take out the highest common factor. Writing \(6x + 12\) as \(2(3x + 6)\) is not wrong arithmetically, but it is not fully factorised because the bracket still has a common factor of \(3\). The complete answer is \(6(x + 2)\).

Revision notes

Finding the common factor

Look at the numbers and the letters separately. Find the highest number dividing into all coefficients and the lowest power of any shared letter.

For \(12x^3 + 8x^2\): the highest common numerical factor is \(4\), and both terms contain at least \(x^2\), so the factor is \(4x^2\).

Writing the bracket

Divide each original term by the factor to find what goes inside the bracket.

Continuing the example: \(12x^3 \div 4x^2 = 3x\) and \(8x^2 \div 4x^2 = 2\), giving \(4x^2(3x + 2)\).

Checking by expanding

Multiply the bracket back out. If you recover the original expression, the factorisation is correct.

\(4x^2(3x + 2) = 12x^3 + 8x^2\), which matches. This check takes seconds and catches incomplete factorising.

Key points

  • Factorising is the reverse of expanding.
  • Take out the highest common factor.
  • Consider numbers and letters separately.
  • Use the lowest power of any shared letter.
  • Divide each term by the factor to fill the bracket.
  • Check by expanding your answer.

Worked examples

Example 1

Factorise \(8x + 12\).

Working

\[\text{HCF of } 8 \text{ and } 12 = 4\]find the highest common factor
\[8x \div 4 = 2x \text{, } 12 \div 4 = 3\]divide each term by the factor
\[4(2x + 3)\]write the factorised form

Example 2

Factorise \(x^2 - 7x\).

Working

\[\text{Both terms contain } x\]x is the common factor
\[x^2 \div x = x \text{, } -7x \div x = -7\]divide each term by x
\[x(x - 7)\]write the factorised form

Example 3

Factorise \(15a^3 + 10a^2\).

Working

\[\text{HCF} = 5a^2\]5 divides both numbers and both terms contain a squared
\[15a^3 \div 5a^2 = 3a \text{, } 10a^2 \div 5a^2 = 2\]divide each term
\[5a^2(3a + 2)\]write the factorised form

Common mistakes

  • Not taking out the highest common factor.

    6x + 12 as 2(3x + 6) is incomplete. The full factorisation is 6(x + 2).

  • Forgetting a letter that is common to all terms.

    In x² + 5x the x is a common factor and should be taken outside.

  • Losing a term from the bracket.

    Each original term must produce a term inside the bracket.

  • Changing a sign when dividing.

    In x² − 7x, dividing −7x by x gives −7, so the bracket is (x − 7).

Exam tips

  • Deal with the numbers and letters separately when finding the factor.
  • Always expand your answer to check it recovers the original.
  • Make sure the bracket has no remaining common factor.
  • Take the lowest power of any letter appearing in all terms.

Key terms

Factorise
To write an expression as a product using brackets.
Common factor
Something that divides into every term.
Highest common factor
The largest factor shared by all terms.
Product
The result of multiplying.

Related topics

Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.