Factoring a Cubic Expression: (f) x^(3)x^(2)y(1)/(3)xy^(2)(1)/(27)y^(3)
This article will guide you through the process of factoring the cubic expression: (f) x^(3)x^(2)y(1)/(3)xy^(2)(1)/(27)y^(3).
Recognizing the Pattern
The given expression exhibits a distinct pattern:
 Descending powers of x: The terms are arranged in descending order of x powers (x³, x², x¹, x⁰).
 Ascending powers of y: The terms are arranged in ascending order of y powers (y⁰, y¹, y², y³).
 Coefficients with specific fractions: The coefficients follow a pattern related to the powers of (1/3) (1, 1/3, 1/9, 1/27).
This pattern suggests that the expression can be factored using the difference of cubes formula:
(a³  b³) = (a  b)(a² + ab + b²)
Applying the Difference of Cubes Formula

Identify 'a' and 'b':
 a = x
 b = (1/3)y

Substitute 'a' and 'b' into the formula:
 (x  (1/3)y) [x² + x(1/3)y + (1/3)²y²]

Simplify the expression:
 (x  (1/3)y) (x² + (1/3)xy + (1/9)y²)
The Factored Expression
Therefore, the factored form of the given cubic expression is:
(x  (1/3)y) (x² + (1/3)xy + (1/9)y²)