Question

what is the completely factored form of \\(x^2 - 16xy + 64y^2\\)?

\\(xy(x - 16 + 64y)\\)
\\(xy(x + 16 + 64y)\\)
\\((x - 8y)(x - 8y)\\)
\\((x + 8y)(x + 8y)\\)

Explanation:

Identify the terms of the trinomial

$$ x^2 - 16xy + 64y^2 $$

Recognize the perfect square trinomial pattern

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

$$ a = x, \quad b = 8y $$

$$ -2ab = -2(x)(8y) = -16xy $$

Write the factored form

$$ (x - 8y)^2 = (x - 8y)(x - 8y) $$

Answer:

• (A) \(xy(x - 16 + 64y)\)
• (B) \(xy(x + 16 + 64y)\)
• (C) \((x - 8y)(x - 8y)\) (Correct answer)
• (D) \((x + 8y)(x + 8y)\)