Factorise 2x³ + 54y³ - 4x - 12y

Factorization is a method of breaking or decomposing any expression or number into smaller expressions or numbers.

Answer: 2x³ + 54y³ - 4x - 12y can be factorised as  2(x + 3y) (x² – 3xy + 9y² – 2)

We will use algebraic identity to factorise 2x³ + 54y³ - 4x - 12y.

Explanation:

From the given expression, 2x³ + 54y³ - 4x - 12y

To factorize the given expression we will perform the following steps.

Step 1: Taking out 2 as a common factor we get  2 (x³ + 27y³ - 2x - 6y)

Step 2: On arranging the like terms we get 2 (x³ + 27y³) – 4 (x + 3y)

Step 3: Using the algebraic identity a³ + b³ = (a + b) (a² – ab + b²), we will factorise (x³ + 27y³). Hence, it can be written as  2 (x + 3y) (x² – 3xy + 9y²) – 4(x + 3y)

Step 4: We can take x + 3y as a common factor as it is present in both the terms.

(x + 3y) [ 2(x² – 3xy + 9y²) – 4 ] 

⇒ 2(x + 3y) [(x² – 3xy + 9y²) – 2]

⇒ 2(x + 3y) (x² – 3xy + 9y² – 2)

Thus, 2x³ + 54y³ - 4x - 12y can be factorised as  2(x + 3y) (x² – 3xy + 9y² – 2)