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(f) X^(3)-x^(2)y(1)/(3)xy^(2)-(1)/(27)y^(3)

2 min read Jun 16, 2024
(f) X^(3)-x^(2)y(1)/(3)xy^(2)-(1)/(27)y^(3)

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

  1. Identify 'a' and 'b':

    • a = x
    • b = (1/3)y

  2. Substitute 'a' and 'b' into the formula:

    • (x - (1/3)y) [x² + x(1/3)y + (1/3)²y²]

  3. 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)

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