Factorize: $16x^4 - y^4$

Analysis

The expression $16x^4 - y^4$ is a difference of two squares. The general formula for the difference of two squares is:

Concept

To factorize this expression, we need to express both terms as perfect squares:

1. $16x^4$ can be written as $(4x^2)^2$
2. $y^4$ can be written as $(y^2)^2$

Step-by-Step Solution

Step 1: Apply the difference of two squares formula $16x^4 - y^4 = (4x^2)^2 - (y^2)^2$ Using $a^2 - b^2 = (a - b)(a + b)$, where $a = 4x^2$ and $b = y^2$: $= (4x^2 - y^2)(4x^2 + y^2)$

Step 2: Factorize the remaining difference of squares Notice that $(4x^2 - y^2)$ can be further factorized because it is also a difference of squares: $4x^2 - y^2 = (2x)^2 - (y)^2$ Applying the formula again: $(2x)^2 - (y)^2 = (2x - y)(2x + y)$

Step 3: Combine all factors Substitute the result from Step 2 back into the expression from Step 1: $= (2x - y)(2x + y)(4x^2 + y^2)$

Conclusion

Comparing our result $(2x - y)(2x + y)(4x^2 + y^2)$ with the given options, it matches Option A.

Exam Tip: Whenever you see a power of 4 (like $x^4$), always check if you can apply the difference of squares rule twice. Also, remember that the sum of squares $(a^2 + b^2)$ cannot be factorized further using real numbers, so leave the $(4x^2 + y^2)$ part as it is!