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

Jasonbradley

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Jun 16, 2024

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

This expression represents the product of a binomial and a trinomial. To expand it, we'll use the distributive property (also known as FOIL).

Understanding the Pattern

The expression (4x^2 + 6xy + 9y^2) is a perfect square trinomial, specifically the square of (2x + 3y). This pattern is helpful for simplifying the expansion.

Expanding the Expression

1. Multiply the first term of the binomial by each term of the trinomial:

   • (2x)(4x^2) = 8x^3
   • (2x)(6xy) = 12x^2y
   • (2x)(9y^2) = 18xy^2

2. Multiply the second term of the binomial by each term of the trinomial:

   • (-3y)(4x^2) = -12x^2y
   • (-3y)(6xy) = -18xy^2
   • (-3y)(9y^2) = -27y^3

3. Combine the results: 8x^3 + 12x^2y + 18xy^2 - 12x^2y - 18xy^2 - 27y^3

4. Simplify by combining like terms: 8x^3 - 27y^3

Final Result

Therefore, the expanded form of (2x - 3y)(4x^2 + 6xy + 9y^2) is 8x^3 - 27y^3.

This result showcases the difference of cubes pattern, which is often seen in algebraic expansions.