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Example 1 Factorize \(81x^3 + 36x^2y + 16y^3\)

A. \((3x)^2(3x + 2y)^2\)

B. \((3x + 2y)^3\)

C. \((3x)^3 + (2y)^3\)

D. \((9x^2 + 4y^2)(9x + 4y)\)

User Birol Efe
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1 Answer

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Final Answer:

The correct factorization for
\(81x^3 + 36x^2y + 16y^3\) is \( (3x + 2y)^3 \)thus option B is correct.

Step-by-step explanation:

To factorize
\(81x^3 + 36x^2y + 16y^3\), we can recognize it as the sum of cubes:
\(a^3 + 3a^2b + 3ab^2 + b^3\), where (a = 3x) and (b = 2y). Applying this pattern, the expression becomes
\((3x)^3 + 3(3x)^2(2y) + 3(3x)(2y)^2 + (2y)^3\). Now, factoring out common terms, we get
\((3x + 2y)^3\).

However, the given options do not match this result. To derive the correct answer, we can recognize that
\(81x^3 + 36x^2y + 16y^3\) is also a sum of squares in the form
\(a^2 + 2ab + b^2\), where (a = 9x) and (b = 2y). Factoring this, we get
\((9x + 4y)^2\), and further factoring the original expression, we obtain
\((9x^2 + 4y^2)(9x + 4y)\). Therefore, the correct answer is D.

In summary, recognizing the expression as both a sum of cubes and a sum of squares allows us to factorize it as
\((9x^2 + 4y^2)(9x + 4y)\), which aligns with option D. This demonstrates the importance of identifying patterns and applying appropriate factorization techniques to arrive at the correct answer.

User NITEMAN
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