Factorising by Grouping (Cambridge International IGCSE Mathematics)

Factorising by grouping

How do I factorise expressions with a common bracket?

• Look at the expression 3x(t + 4) + 2(t + 4)

  • Both terms have a common bracket, (t + 4)

  • The whole bracket, (t + 4), can be "taken out" like a common factor:

    • (t + 4)(3x + 2)

• This is like factorising 3xy + 2y to get y(3*x *+ 2)

  • y represents (t + 4) above

How do I factorise by grouping?

• Some questions may require you to form a common bracket yourself

  • For example xy + 3x + 5y + 15

    • The first two terms have a common factor of x
    • The second two terms have a common factor of 5

  • Factorising fully the first pair of terms, and the last pair of terms:

    • x(y + 3) + 5(y + 3)

  • You can now spot a common bracket of (y + 3)

    • (y + 3)(x + 5)

• This is called factorising by grouping

Does it matter what order I group in?

• You can often rearrange terms to factorise in a different order

  • Rewriting the same example, xy + 3x + 5y + 15, but in a different order:

    • xy + 5y + 3x + 15
    • The first pair of terms have a common factor of y
    • The second pair of terms have a common factor of 3

  • Factorising gives y(x + 5) + 3(x + 5)

    • You can now spot a common bracket, this time of (x + 5)

  • (x+5)(y+3)

    • This gives the same result as found previously

• Some rearrangements cannot be factorised as "first pair" then "second pair"

  • For example, rewriting the above example as xy + 15 + 3x + 5y

Exam Hint Once you have factorised something, expand it by hand to check your answer is correct.

Worked Example Factorise ab + 3*b + 2a *+ 6.

Answer:

Method 1

*Notice that ab and 3b have a common factor of *b *Notice that 2**a *and 6 have a common factor of 2

Factorise the first two terms, using b as a common factor

** b(a + 3) + 2a **+ 6 *

Factorise the second two terms, using 2 as a common factor

** b(a + 3) + 2( a **+ 3) *

*(**a **+ 3) is a common bracket * We can now factorise out the bracket (a + 3)

Final answer: (a + 3)(b + 2)

Method 2

*Notice that ab and 2a have a common factor of *a Notice that 3b and 6 have a common factor of 3

Rewrite the expression, grouping these terms together

**ab + 2a + 3b *+ 6

*Factorise the first two terms, using a as a common factor *

a(b + 2) + 3b + 6*

Factorise the second two terms, using 3 as a common factor

a(b + 2) + 3(b + 2) *

(b+ 2) is a common bracket

• We can now factorise out the bracket (b + 2)*

Final answer: (b + 2)(a + 3)