(x+2y)(x^2-2xy+4y^2)-(2y-3x)(4y^2+6xy+9x^2)

Jasonbradley

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Jun 16, 2024

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Simplifying the Expression: (x+2y)(x^2-2xy+4y^2)-(2y-3x)(4y^2+6xy+9x^2)

This expression involves the multiplication of two sets of binomials. To simplify it, we'll use the distributive property and some algebraic identities.

Understanding the Patterns

• First Set: The first set of binomials, (x+2y)(x^2-2xy+4y^2), resembles the pattern of a sum of cubes: (a+b)(a^2-ab+b^2) = a^3 + b^3.
• Second Set: The second set, (2y-3x)(4y^2+6xy+9x^2), resembles the pattern of a difference of cubes: (a-b)(a^2+ab+b^2) = a^3 - b^3.

Applying the Identities

1. Expanding the first set:

   • (x+2y)(x^2-2xy+4y^2) = x^3 + (2y)^3 = x^3 + 8y^3

2. Expanding the second set:

   • (2y-3x)(4y^2+6xy+9x^2) = (2y)^3 - (3x)^3 = 8y^3 - 27x^3

Combining the Results

Now we have: (x^3 + 8y^3) - (8y^3 - 27x^3)

Simplifying this, we get:

x^3 + 8y^3 - 8y^3 + 27x^3 = 28x^3

Therefore, the simplified form of the expression (x+2y)(x^2-2xy+4y^2)-(2y-3x)(4y^2+6xy+9x^2) is 28x^3.