- Recognize the expression \((2x^2 - 3y^2)(4x^4 + 6x^2y^2 + 9y^4)\) as the algebraic identity for the difference of cubes: \((a - b)(a^2 + ab + b^2) = a^3 - b^3\).
- Identify \(a = 2x^2\) and \(b = 3y^2\) to confirm the pattern.
- Calculate the resulting cubes: \(a^3 = (2x^2)^3 = 8x^6\) and \(b^3 = (3y^2)^3 = 27y^6\).
- Conclude that the expression simplifies to \(\boxed{8x^6 - 27y^6}\).
Explanation
- Analyzing the Expression Structure
To simplify the expression \((2x^2 - 3y^2)(4x^4 + 6x^2y^2 + 9y^4)\), we analyze its structure. The expression consists of a binomial multiplied by a trinomial, which often corresponds to a known algebraic identity. Recall the difference of cubes formula:
\[(a - b)(a^2 + ab + b^2) = a^3 - b^3\]
By comparing our expression to this identity, we let \(a = 2x^2\) and \(b = 3y^2\). We now verify if the trinomial \(4x^4 + 6x^2y^2 + 9y^4\) matches the form \(a^2 + ab + b^2\).
- Verifying the Identity Components
First, we calculate the components of the identity:
- \(a^2 = (2x^2)^2 = 4x^4\)
- \(ab = (2x^2)(3y^2) = 6x^2y^2\)
- \(b^2 = (3y^2)^2 = 9y^4\)
The expression \((2x^2 - 3y^2)(4x^4 + 6x^2y^2 + 9y^4)\) exactly matches the form \((a - b)(a^2 + ab + b^2)\). Therefore, according to the difference of cubes identity, the result is equivalent to \(a^3 - b^3\).
- Applying the Identity and Conclusion
Now, we substitute \(a = 2x^2\) and \(b = 3y^2\) back into the formula \(a^3 - b^3\):
\[a^3 = (2x^2)^3 = 8x^6\]
\[b^3 = (3y^2)^3 = 27y^6\]
Thus, the expression simplifies to \(8x^6 - 27y^6\). Comparing this result with the given choices, we see that it corresponds to the fourth option.
Examples
Algebraic identities like the difference of cubes are foundational in fields such as engineering, physics, and computer science. For example, in structural analysis, these identities are used to simplify complex polynomial models of stress and strain distribution across materials. Efficiently simplifying such expressions saves computation time in simulations and allows engineers to isolate critical variables, leading to safer and more durable designs.