Everything Maths Grade 10

Worked example 14: Factorising a difference of two cubes

QUESTION

Factorise:a^3 1.

SOLUTION

Step 1: Take the cube root of terms that are perfect cubes

We are working with the difference of two cubes. We know thatx^3 y^3 = (xy)

(

x^2 +xy+y^2

)

, so we
need to identifyxandy.

We start by noting that^3

p

a^3 =aand^3

p
1 = 1. These give the terms in the first bracket. This also tells us that
x=aandy= 1.

Step 2: Find the three terms in the second bracket

We can replacexandyin the factorised form of the expression for the difference of two cubes withaand 1.
Doing so we get the second bracket: (
a^3 1

)

= (a1)

(

a^2 +a+ 1

)

Step 3: Expand the brackets to check that the expression has been correctly factorised

(a1)

(

a^2 +a+ 1

)

=a

(

a^2 +a+ 1

)

1

(

a^2 +a+ 1

)

=a^3 +a^2 +aa^2 a 1

=a^3  1

Worked example 15: Factorising a sum of two cubes

QUESTION

Factorise:x^3 + 8.

SOLUTION

Step 1: Take the cube root of terms that are perfect cubes

We are working with the sum of two cubes. We know thatx^3 +y^3 = (x+y)

(

x^2 xy+y^2

)

, so we need to
identifyxandy.

We start by noting that^3

p

x^3 =xand^3

p
8 = 2. These give the terms in the first bracket. This also tells us that
x=xandy= 2.

Step 2: Find the three terms in the second bracket

We can replacexandyin the factorised form of the expression for the sum of two cubes withxand 2. Doing
so we get the second bracket: (
x^3 + 8

)

= (x+ 2)

(

x^2  2 x+ 4

)

Step 3: Expand the brackets to check that the expression has been correctly factorised

(x+ 2)

(

x^2  2 x+ 4

)

=x

(

x^2  2 x+ 4

)

+ 2

(

x^2  2 x+ 4

)

=x^3  2 x^2 + 4x+ 2x^2  4 x+ 8

=x^3 + 8

28 1.7. Factorisation