Question

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

Answer 1

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.