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Factorization

Factorization

Factorising: common factors and the difference of two squares

Two Terms Core + Extended 6 min read

On this page
  1. In this note
  2. Formula bank
  3. What factorising means
  4. Taking out the highest common factor
  5. The difference of two squares
  6. Ways they test it
  7. Mistakes to avoid
  8. Summary
  9. Practice
  10. Can I do this?
  11. Quiz

In this note

Syllabus 2.2: factorising two-term expressions
  • 2.2Take out the highest common factor (numbers, letters and powers) and factorise fully.
  • 2.2Factorise a difference of two squares, including after taking out a common factor.
  • 2.2Factorise when the common factor is a whole bracket, e.g. \(3(k+g)^2-5(k+g)\).

Before you start: expanding brackets and HCF. Next: Factorising quadratics.

Formula bank

Common factorLearn it\(ab+ac=a(b+c)\)a = the highest common factor of every term.Use it when every term shares a number or a letter.
Difference of two squaresLearn it\(a^{2}-b^{2}=(a+b)(a-b)\)a² and b² are both perfect squares; there is a MINUS between them.Use it when two terms, a minus, both squares.

What factorising means

Factorising is expanding backwards: you write the expression as things multiplied together. \(4x(2x-3)\) expands to \(8x^{2}-12x\), so \(8x^{2}-12x\) factorises to \(4x(2x-3)\). Always check by expanding your answer.

Taking out the highest common factor

General methodHow to take out a common factor

  1. Find the biggest number that divides every number in front.
  2. For each letter that is in EVERY term, take its lowest power.
  3. Write that HCF outside a bracket. Inside, divide each term by it.
  4. Check: expand back, and make sure nothing common is left inside (that is what 'fully' means).

Example Factorise fully \(15a^{2}b+10ab^{2}\).

HCF of 15 and 10 is 5; lowest powers \(a\) and \(b\): HCF \(5ab\).

  1. \(15a^{2}b\div5ab=3a\)
  2. \(10ab^{2}\div5ab=2b\)

Answer \(5ab(3a+2b)\)

A whole bracket can be the common factor. Treat it like one letter: \(3(k+g)^{2}-5(k+g)=(k+g)\big(3(k+g)-5\big)=(k+g)(3k+3g-5)\).

Try it yourself

1
Factorise fully \(12x-18\).
Show answer

Answer \(6(2x-3)\)

2
Factorise fully \(10x^{2}-15x\).
Show answer

Answer \(5x(2x-3)\)

3
Factorise fully \(14xy-7y^{2}\).
Show answer

Answer \(7y(2x-y)\)

The difference of two squares

Expand \((a+b)(a-b)\): the middle terms \(-ab+ab\) cancel and you get \(a^{2}-b^{2}\). So a square minus a square always factorises this way. A SUM of squares, like \(x^{2}+49\), does not factorise.

General methodHow to factorise a difference of two squares

  1. Take out any common factor first (it often turns the numbers into squares).
  2. Write each term as something squared: \(9a^{2}=(3a)^{2}\), \(25=5^{2}\).
  3. Write \((\text{first}+\text{second})(\text{first}-\text{second})\).
  4. Check each bracket: can it be factorised again?

Example Factorise fully \(3x^{2}-48y^{2}\).

48 isn't a square, so take out 3 first.

  1. \(3(x^{2}-16y^{2})\)
  2. \(x^{2}-16y^{2}=(x)^{2}-(4y)^{2}\)
  3. \(=3(x+4y)(x-4y)\)

Answer \(3(x+4y)(x-4y)\)

Try it yourself

1
Factorise \(x^{2}-100\).
Show answer

Answer \((x+10)(x-10)\)

2
Factorise \(4x^{2}-1\).
Show answer

Answer \((2x+1)(2x-1)\)

3
Work out \(53^{2}-47^{2}\) without a calculator.
Hint
\((53+47)(53-47)\).
Show answer

Answer 600

Ways they test it

Type 1 Factorise completely (common factor)

"Factorise completely \(18x^{3}y-24x^{2}y^{2}\)." (2 marks)

Spot it one bracket answer; 'completely' means the biggest HCF.

  1. HCF of the numbers.
  2. Lowest power of each letter.
  3. Divide each term.

For example \(18x^{3}y-24x^{2}y^{2}\)

  1. HCF \(6x^{2}y\)

Answer \(6x^{2}y(3x-4y)\)

Type 2 Difference of two squares

"Factorise \(9a^{2}-25\)." (1 mark)

Spot it two terms, a minus, both squares.

  1. Write as \((3a)^{2}-5^{2}\).
  2. (sum)(difference).

For example \(9a^{2}-25\)

    Answer \((3a+5)(3a-5)\)

    Type 3 Common factor, then DOTS

    "Factorise fully \(50x^{2}-8y^{2}\)." (2 marks)

    Spot it the numbers aren't squares until you take something out.

    1. Take out the HCF.
    2. Then DOTS on the bracket.

    For example \(50x^{2}-8y^{2}\)

    1. \(2(25x^{2}-4y^{2})\)

    Answer \(2(5x+2y)(5x-2y)\)

    Type 4 A hidden bracket

    "Factorise \(x^{2}(a-b)+4(b-a)\)." (3 marks)

    Spot it brackets that differ only by order: \((b-a)=-(a-b)\).

    1. Rewrite \(+4(b-a)\) as \(-4(a-b)\).
    2. Take out \((a-b)\).
    3. DOTS on \(x^{2}-4\).

    For example \(x^{2}(a-b)-4(a-b)\)

    1. \((a-b)(x^{2}-4)\)

    Answer \((a-b)(x+2)(x-2)\)

    Type 5 Quick arithmetic

    "Work out \(101^{2}-99^{2}\) without a calculator."

    Spot it two squares of big numbers with a minus.

    1. (sum) × (difference).

    For example \(101^{2}-99^{2}\)

    1. \(200\times2\)

    Answer 400

    Mistakes to avoid

    • ✗ \(x^{2}+49=(x+7)(x-7)\)✓ it doesn't factoriseDOTS needs a minus.
    • ✗ \(8x^{2}-12x=2(4x^{2}-6x)\)✓ \(4x(2x-3)\)Not fully: 2 and x are still common inside.
    • ✗ \(3x^{2}-48y^{2}=(\sqrt3x+\ldots)\)✓ \(3(x+4y)(x-4y)\)Take out the common factor first.
    • ✗ \(x^{4}-16=(x^{2}+4)(x^{2}-4)\)✓ \((x^{2}+4)(x+2)(x-2)\)\(x^{2}-4\) is another DOTS.

    Summary

    • Factorising = expanding backwards; check by expanding.
    • Common factor: HCF of the numbers × lowest power of each shared letter.
    • \(a^{2}-b^{2}=(a+b)(a-b)\); a sum of squares does not factorise.
    • Always take out the common factor FIRST, then look for DOTS.

    If you see …

    • "factorise completely / fully"biggest HCF, then check every bracket again.
    • two terms with a minusare they both squares (after taking out an HCF)?

    Practice

    Practice: easy, medium, hard

    Easy

    1
    Factorise \(6x+9\).
    Show answer

    Answer \(3(2x+3)\)

    2
    Factorise fully \(8x^{2}-12x\).
    Hint
    HCF of 8 and 12 is 4; both terms have x.
    Show answer

    Answer \(4x(2x-3)\)

    3
    Factorise \(x^{2}-49\).
    Show answer

    Answer \((x+7)(x-7)\)

    Medium

    1
    Factorise \(9a^{2}-25\).
    Hint
    \(9a^{2}=(3a)^{2}\).
    Show answer

    Answer \((3a+5)(3a-5)\)

    2
    Factorise completely \(2x^{2}-18\).
    Guide me

    What is it asking? a product

    What do we know? 18 isn't a square; 2 is common

    Plan take out 2, then DOTS

    Show working
    1. \(2(x^{2}-9)\)
    2. \(2(x+3)(x-3)\)

    Answer \(2(x+3)(x-3)\)

    3
    Factorise completely \(24x^{2}-9xy\).
    Guide me

    What is it asking? a product

    What do we know? 3 and x are in both terms

    Plan HCF 3x

    Show working
    1. \(24x^{2}\div3x=8x\)
    2. \(9xy\div3x=3y\)

    Answer \(3x(8x-3y)\)

    Hard

    1
    Factorise completely \(20a^{3}-45a\).
    Guide me

    What is it asking? a product

    What do we know? HCF \(5a\)

    Plan then \(4a^{2}-9\) is DOTS

    Show working
    1. \(5a(4a^{2}-9)\)
    2. \(5a(2a+3)(2a-3)\)

    Answer \(5a(2a+3)(2a-3)\)

    PartAnswer / workingMarksGuidance
    \(5a(4a^{2}-9)\)M1
    \(5a(2a+3)(2a-3)\)A1
    Total: 2 marks
    2
    Factorise completely \(63x^{2}-28y^{2}\).
    Guide me

    What is it asking? a product

    What do we know? 7 is common

    Plan then 9x² − 4y² is DOTS

    Show working
    1. \(7(9x^{2}-4y^{2})\)
    2. \(7(3x+2y)(3x-2y)\)

    Answer \(7(3x+2y)(3x-2y)\)

    PartAnswer / workingMarksGuidance
    \(7(9x^{2}-4y^{2})\)M1
    \(7(3x+2y)(3x-2y)\)A1
    Total: 2 marks
    3
    Factorise completely \(x^{4}-16\).
    Guide me

    What is it asking? a product

    What do we know? \(x^{4}=(x^{2})^{2}\), \(16=4^{2}\)

    Plan DOTS twice

    Show working
    1. \((x^{2}+4)(x^{2}-4)\)
    2. \((x^{2}+4)(x+2)(x-2)\)

    Answer \((x^{2}+4)(x+2)(x-2)\)

    PartAnswer / workingMarksGuidance
    \((x^{2}+4)(x^{2}-4)\)M1
    \((x^{2}+4)(x+2)(x-2)\)A1
    Total: 2 marks

    Can I do this?

    Anything unticked: revise that section first. All ticked: take the quiz.

    Quiz

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