Factorising 2x^3 - xy^2 - y^3

To factorise the expression 2x^3 - xy^2 - y^3, we will look for common factors and then use various factorisation techniques such as grouping, difference of cubes, or the quadratic formula if necessary.

Step 1: Common Factors

First, let's check if there are any common factors among the terms in the expression. In this case, there are no common factors.

Step 2: Grouping

Since there are no common factors, we can try grouping the terms in pairs to see if any factorisation pattern emerges.

Grouping the first two terms (2x^3 - xy^2) and the last term (-y^3), we get:

2x^3 - xy^2 - y^3 = (2x^3 - xy^2) - y^3

Step 3: Difference of Cubes

Now, let's take a closer look at the first group (2x^3 - xy^2). It appears to be a difference of cubes pattern. The general formula for a difference of cubes is:

a^3 - b^3 = (a - b)(a^2 + ab + b^2)

Applying this formula to the first group, where a = 2x and b = y, we have:

(2x^3 - xy^2) = (2x - y)(4x^2 + 2xy + y^2)

Step 4: Final Factorisation

Now, we have:

2x^3 - xy^2 - y^3 = (2x - y)(4x^2 + 2xy + y^2) - y^3

We can simplify this further by distributing the -y^3 to the terms inside the parentheses:

2x^3 - xy^2 - y^3 = (2x - y)(4x^2 + 2xy + y^2) - y^3
= (2x - y)(4x^2 + 2xy + y^2) - y^3

Therefore, the fully factorised form of the expression 2x^3 - xy^2 - y^3 is (2x - y)(4x^2 + 2xy + y^2) - y^3.