Final Answer: $ oxed{(2x - 3y)(2x + 3y)(4x^2 + 9y^2)} $ — Simplified, Factored, and Expanded Form Explained

Unlock the Power of Algebraic Identity: Final Answer & Full Breakdown

Understanding the Context

When simplifying complex expressions, mastering algebraic identities can dramatically improve clarity and computation efficiency. One powerful expression often encountered is:

$$
oxed{(2x - 3y)(2x + 3y)(4x^2 + 9y^2)}
$$

At first glance, this triple product may seem intimidating. But with strategic factoring and recognition of standard forms, we unlock a beautifully clean and efficient result.

Final answer: $ \boxed{(2x - 3y)(2x + 3y)(4x^2 + 9y^2)} $

Key Insights

Step 1: Recognize and Apply the Difference of Squares

Observe that the first two factors, $ (2x - 3y) $ and $ (2x + 3y) $, form a classic difference of squares:

$$
(a - b)(a + b) = a^2 - b^2
$$

Let $ a = 2x $ and $ b = 3y $. Then:

$$
(2x - 3y)(2x + 3y) = (2x)^2 - (3y)^2 = 4x^2 - 9y^2
$$

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Now use $ p(1) $: $ 0 + 2 + 1 + d = 3 \Rightarrow d = 0 $.
Thus, $ p(x) = 2x^2 + x $. Then $ p(0) = 0 $.
But wait—this is quadratic, but the problem states $ p(x) $ is cubic. Contradiction? Not necessarily—$ a = 0 $ is allowed if the cubic coefficient is zero, but the problem says cubic, implying degree exactly 3. So we must assume it's cubic, but our solution gives $ a = 0 $. That means the data fits a quadratic, but we are told it's cubic. So either the model is misclassified, or we must accept the interpolating polynomial, regardless of degree. Since the interpolation yields a unique cubic (degree at most 3), and we found $ a = 0 $, the polynomial is actually quadratic. But the problem says cubic, so we must reconsider.

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Final Thoughts

Now the expression simplifies elegantly to:

$$
(4x^2 - 9y^2)(4x^2 + 9y^2)
$$

Step 2: Apply the Difference of Squares Again

Notice that the remaining expression is again a difference of squares:

$$
(4x^2 - 9y^2)(4x^2 + 9y^2) = (4x^2)^2 - (9y^2)^2 = 16x^4 - 81y^4
$$

This final result — $ 16x^4 - 81y^4 $ — is a simple quartic difference of squares, showcasing how repeated application of fundamental identities leads to simplified algebra.

Why This Matters: Key Takeaways

  • Efficiency: Recognizing patterns like difference of squares saves time and reduces error.
  • Factoring Power: Mastery of identities transforms complicated expressions into compact forms.
  • Mathematical Beauty: The step-by-step decomposition mirrors the logical structure underlying algebraic reasoning.
  • Real-World Applications: Such simplifications appear in calculus, optimization, and physics equations involving quadratic forms.