24x + 20 = 4(□x + □) Fill in each blank

6, 5. The H.C.F. of 24 and 20 is 4: 24x ÷ 4 = 6x and 20 ÷ 4 = 5.

xy + x + 5y + 5 = x(y + □) + □(y + 1) Fill in each blank

1, 5. xy + x = x(y + 1) and 5y + 5 = 5(y + 1): the two brackets match.

x^3 − 216 can be factorised as (x + 6)(x^2 − 6x + 36). True or false?

False. (x + 6)(x^2 − 6x + 36) = x^3 + 216, a sum of cubes; x^3 − 216 factorises as (x − 6)(x^2 + 6x + 36).

Algebraic expressions · Ages 13–14

Factorisation

From “Taking out a common factor” to “Factorising sums and differences of cubes”, step by step, each skill with its methods, worked examples and answers.

Skills and methods in this topic, easy to hard

Taking out a common factorThe coefficients' H.C.F., each letter to its lowest index
Method6x² − 9x = 3x(2x − 3)
Another way: Expand to check3x(2x − 3) = 6x² − 9x
Factorising a difference of two squaresTwo squares with a minus between; a common factor first
Method9x² − 16 = (3x + 4)(3x − 4)
Another way: Expand to check(3x + 4)(3x − 4) = 9x² − 16
Factorising perfect squaresFirst and last are squares, the middle twice their product
Methodx² − 10x + 25 = (x − 5)²
Another way: Expand to check(x − 5)² = x² − 10x + 25
Factorising by groupingPairs, a factor out of each, the brackets must match
Methodxy + 3x + 2y + 6 = (y + 3)(x + 2)
Another way: Group the other way(xy + 2y) + (3x + 6) = y(x + 2) + 3(x + 2)
The cross method: leading coefficient 1Two numbers: product c, sum b
Methodx² + 7x + 12 = (x + 3)(x + 4)
Another way: Draw the cross3x + (−5x) = −2x
The cross method: other leading coefficientsSplit a and c; the cross products must add to b
Method2x² + 7x + 3 = (2x + 1)(x + 3)
Another way: Split the middle term2x² + 6x + x + 3 = 2x(x + 3) + (x + 3)
Factorising sums and differences of cubesCommon factor first, then a³ ± b³; work out a², ab, b² whole
Method8x³ − 27y³ = (2x − 3y)(4x² + 6xy + 9y²)
Another way: Check with a valuex = 1: 2 − 54 = −52, 2 × (−2) × 13 = −52

Worked examples

From different skills in this topic. Have a go first, then open each one for the answer and the working.

Example 1Taking out a common factor →
24x + 20 = 4(x + )
Fill in each blank
Show the answer and working
Answer: 6, 5
The H.C.F. of 24 and 20 is 4: 24x ÷ 4 = 6x and 20 ÷ 4 = 5.
Example 2Factorising by grouping →
xy + x + 5y + 5 = x(y + ) + (y + 1)
Fill in each blank
Show the answer and working
Answer: 1, 5
xy + x = x(y + 1) and 5y + 5 = 5(y + 1): the two brackets match.
Example 3Factorising sums and differences of cubes →
x³ − 216 can be factorised as (x + 6)(x² − 6x + 36).
True or false?
Show the answer and working
Answer: False
(x + 6)(x² − 6x + 36) = x³ + 216, a sum of cubes; x³ − 216 factorises as (x − 6)(x² + 6x + 36).

Free worksheet: FactorisationWorking out, multiple choice and word problems, with answers and working, to print or save as a PDF

VIP practice notes and papers

Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.

Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
Practice paper: Set out in parts like a school paper, marked, with answers and full working

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