Practice Maths

Factorising with Common Factors

Key Ideas

Key Terms

factorising
Rewriting an expression as a product of factors — the reverse of expanding.
highest common factor (HCF)
The largest factor that divides every term in an expression — extracted outside the brackets when factorising.
grouping
A factorising technique for four-term expressions where pairs of terms are grouped and each pair's HCF is extracted.
Method Steps Example
HCF factorisingFind HCF of all terms; divide each term by HCF6x + 9 = 3(2x + 3)
Variable HCFInclude lowest power of each shared variablex2 + 5x = x(x + 5)
Grouping (4 terms)Group into pairs, factorise each, extract common bracketax + ay + bx + by = (a + b)(x + y)
Hot Tip After factorising, always expand your answer to verify you get back to the original. Also check that the terms inside the bracket have no further common factor — if they do, you haven’t taken out the full HCF.

Worked Example

Question: Factorise 6x2y − 9xy2.

Step 1: HCF of 6 and 9 = 3.   Step 2: Both have x and y; lowest powers are x1y1.

Step 3: HCF = 3xy.   Divide: 6x2y ÷ 3xy = 2x;   9xy2 ÷ 3xy = 3y.

Result: 3xy(2x − 3y)   Check: 3xy × 2x − 3xy × 3y = 6x2y − 9xy2 ✓

Factorising as the Reverse of Expanding

Expanding turns a factorised expression into an expanded one: 3(x + 4) = 3x + 12. Factorising is the reverse: starting from 3x + 12, rewrite it as 3(x + 4). Factorising is one of the most important skills in algebra because it is used to simplify fractions, solve equations, and analyse graphs.

Finding the Highest Common Factor (HCF)

The HCF is the largest factor that divides every term in the expression. To find it:

1. Find the HCF of the numerical coefficients. For 12x2 + 8x: HCF of 12 and 8 is 4.

2. Find the HCF of the variable parts. x2 and x both contain x, so take the lowest power: x.

3. HCF = 4x. So: 12x2 + 8x = 4x(3x + 2).

More examples: 6a2b + 9ab2 = 3ab(2a + 3b). 15x3 − 10x2 + 5x = 5x(3x2 − 2x + 1).

Factorising with Negative Common Factors

Sometimes it is convenient to take out a negative common factor, particularly when the leading term has a negative coefficient. Taking out −1 (or a negative HCF) reverses the signs of every term inside the bracket.

Example: −3x + 6 = −3(x − 2). Check: −3(x − 2) = −3x + 6. Correct.

Example: −2x2 − 4x = −2x(x + 2). Check: −2x(x + 2) = −2x2 − 4x. Correct.

Checking Your Answer

Always verify your factorisation by expanding the brackets. If you get back to the original expression, you are correct. If not, recheck your HCF or the signs inside the bracket.

The factorised form should have no common factor remaining inside the bracket. If you can still take out a factor from inside, you have not fully factorised — you chose a factor smaller than the HCF.

Strategy tip: To check you have the HCF and not just any common factor, ask: "Is there any factor (other than 1) that divides every term inside the bracket?" If yes, your answer is incomplete. For example, 4(2x + 6) is a valid factorisation of 8x + 24 but not the fully factorised form — the HCF is 8, giving 8(x + 3). Always aim for the fully factorised form.

Mastery Practice

  1. Factorise by taking out the highest common numerical factor. Fluency

     ExpressionHCFFactorised Form
    (a)6x + 9  
    (b)12y − 8  
    (c)15a + 20  
    (d)18m − 24n  
    (e)14p + 21q − 7  
    (f)30x − 45y + 15  
    (g)8a + 12b − 16c  
    (h)100x − 75y  
    (i)x2 + 5x  
    (j)3y2 − 9y  

  2. Factorise by taking out the full HCF, including variables. Fluency

     ExpressionFactorised Form
    (a)4a3 + 6a2 
    (b)8m2n − 12mn2 
    (c)5p3 − 15p2 + 10p 
    (d)6x2y + 4xy2 − 2xy 
    (e)9a2b2 − 3a2b 
    (f)16m3n2 − 24m2n3 
    (g)12x3 − 18x2 + 6x 
    (h)20a2b − 30ab2 + 10ab 

  3. Factorise these four-term expressions using the grouping method. Fluency

     ExpressionFactorised Form
    (a)ax + 3a + bx + 3b 
    (b)mx − 2m + nx − 2n 
    (c)xy + 2x + 3y + 6 
    (d)pq − 4p − 3q + 12 
    (e)6ab + 9a − 4b − 6 
    (f)2x2 + 6x + 5x + 15 
    (g)3x2 + 12x + 2x + 8 
    (h)ab + ac + b2 + bc 

  4. True or False? Write T or F and fix any false statements. Fluency

     StatementT / F
    (a)12x2 + 8x fully factorised is 4(3x2 + 2x). 
    (b)The HCF of 6x2 and 9x is 3x. 
    (c)Factorising is the reverse process of expanding. 
    (d)−3y2 + 9y fully factorised is 3y(−y + 3). 

  5. Area and perimeter with factorising. Understanding

    Two interpretations. The area of a rectangle is given by A = 6x2 + 9x.
    1. Factorise the expression to find two possible expressions for the length and width.
    2. If x = 5, calculate the area using both the original and the factorised form. Do you get the same answer?
    3. Write an expression for the perimeter of the rectangle in terms of your length and width expressions.

  6. Factorising with a binomial common factor. Understanding

    Spot the common bracket. Sometimes a whole bracket acts as the common factor.
    1. Factorise 6a(a + 3) − 4(a + 3). (Hint: treat (a + 3) as the common factor.)
    2. Factorise x(x − 2) + 5(x − 2).
    3. Verify your answer to part (a) by expanding and confirming you get back to the original expression.

  7. Fully factorise — check for a further common factor. Understanding

    Go all the way. An expression is fully factorised only when no further common factor exists.
    1. Factorise −4p3 − 8p2 completely.
    2. A student writes: 20a2b − 30ab2 + 10ab = 10(2a2b − 3ab2 + ab). Is this fully factorised? If not, complete the factorisation.
    3. Factorise 6a2b2 − 9a2b + 3ab completely.

  8. Proof by factorising. Understanding

    Always and never. Factorising can help us prove things are always true.
    1. Show that n2 + n is always even for any positive integer n by factorising and explaining why the result is always divisible by 2.
    2. Show that n2 − n is also always even for any positive integer n.

  9. Perimeter, profit and factorising. Problem Solving

    Real-world expressions. A manufacturer makes n items. Total revenue is R = 12n2 + 8n dollars and total cost is C = 4n dollars.
    1. Write an expression for the profit P = R − C.
    2. Factorise the profit expression completely.
    3. Find the profit when n = 50 items using your factorised form.

  10. Shape perimeter investigation. Problem Solving

    Two ways to calculate. The perimeter of a shape is given by P = 14a − 21b + 7.
    1. Factorise this expression completely.
    2. Using a = 5 and b = 2, calculate the perimeter using the original expression.
    3. Now calculate the perimeter using the factorised form. Confirm you get the same answer.
    4. For what values of a and b would the perimeter equal zero? Discuss whether this is physically meaningful.
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