Question

The factor of \( a^4 b^4 - 16c^4 \) is:

• A. \( (a^2 b^2 - 4c^2)(ab + 2c)(ab - 4c) \)
• B. \( (a^2 b^2 - 4c^2)(ab + 2c)(ab + 4c) \)
• C. \( (a^2 b^2 + 4c^2)(ab + 2c)^2 \)
• D. \( (a^2 b^2 - 4c^2)(ab + 2c)^2 \)

Hint:

To factor expressions of the form \( a^4 - b^4 \), use the difference of squares technique, and continue factoring each term as needed.

Answer 1

The Correct Option is B

We begin by factoring the given expression \( a^4 b^4 - 16c^4 \). This is a difference of squares: \[ a^4 b^4 - 16c^4 = (a^2 b^2 + 4c^2)(a^2 b^2 - 4c^2) \] Next, factor \( a^2 b^2 - 4c^2 \) as another difference of squares: \[ a^2 b^2 - 4c^2 = (ab + 2c)(ab - 2c) \] Thus, the complete factorization is: \[ a^4 b^4 - 16c^4 = (a^2 b^2 + 4c^2)(ab + 2c)(ab - 4c) \] Therefore, the correct answer is \( (a^2 b^2 - 4c^2)(ab + 2c)(ab + 4c) \).