Question

which expression is equivalent to 16a⁴ - b⁴?
a. (2a + b)(2a - b)(4a² + b²)
b. (2a + b)(2a + b)(2a - b)(2a - b)
c. (2a - b)(2a - b)(2a - b)(2a - b)
d. (2a - b)(2a - b)(4a² + b²)

Explanation:

Step1: Recognize the difference of squares

The expression \(16a^4 - b^4\) can be written as \((4a^2)^2 - (b^2)^2\), which is a difference of squares. The formula for the difference of squares is \(x^2 - y^2=(x + y)(x - y)\). Here, \(x = 4a^2\) and \(y = b^2\), so we have:
\((4a^2)^2 - (b^2)^2=(4a^2 + b^2)(4a^2 - b^2)\)

Step2: Factor \(4a^2 - b^2\) further

The expression \(4a^2 - b^2\) is also a difference of squares, where \(x = 2a\) and \(y = b\). Using the difference of squares formula again:
\(4a^2 - b^2=(2a + b)(2a - b)\)

Step3: Combine the factors

Substituting the factored form of \(4a^2 - b^2\) back into the previous expression, we get:
\(16a^4 - b^4=(4a^2 + b^2)(2a + b)(2a - b)=(2a + b)(2a - b)(4a^2 + b^2)\)

Answer:

A. \((2a + b)(2a - b)(4a^2 + b^2)\)