Factor the difference of squares: \( 16a^{4} - b^{2} \)

To factor the expression \( 16a^{4} - b^{2} \) as a difference of squares, follow these steps: ### Step 1: Identify the Perfect Squares First, recognize that both terms in the expression are perfect squares: - \( 16a^{4} \) can be written as \( (4a^{2})^{2} \) - \( b^{2} \) is already a perfect square ### Step 2: Apply the Difference of Squares Formula The difference of squares formula is: \[ A^{2} - B^{2} = (A + B)(A - B) \] Apply this formula to the expression: \[ 16a^{4} - b^{2} = (4a^{2})^{2} - (b)^{2} = (4a^{2} + b)(4a^{2} - b) \] ### Step 3: Check for Further Factoring Examine each factor to see if it can be factored further: 1. **\( 4a^{2} + b \):** This is a sum of squares, which cannot be factored further using real numbers. 2. **\( 4a^{2} - b \):** This is a difference of squares only if \( b \) is a perfect square. Since \( b \) is a variable and not necessarily a square term, it cannot be factored further in general. ### Final Factored Form \[ 16a^{4} - b^{2} = (4a^{2} + b)(4a^{2} - b) \] This is the fully factored form of the given expression.