Factor \( x^{4}-16 \) \( \left(x^{2}+4\right)\left(x^{2}-4\right) \) None of these \( \left(x^{2}+4\right)(x-2)^{2} \) \( (x-2)(x+2)\left(x^{2}+4\right) \)

To factor \( x^{4}-16 \), you can recognize it as a difference of squares, which allows us to rewrite it as \( (x^{2})^{2} - (4)^{2} \). Applying the difference of squares formula \( a^2 - b^2 = (a - b)(a + b) \), we get: \[ (x^{2} - 4)(x^{2} + 4) \] Next, notice that \( x^{2} - 4 \) can be further factored as a difference of squares itself: \[ (x - 2)(x + 2)(x^{2} + 4) \] So, the completely factored form is: \[ (x - 2)(x + 2)(x^{2} + 4) \] Now, let's explore some interesting links! First, the concept of factoring polynomials using the difference of squares dates back to ancient mathematics. The method was used by mathematicians such as the Greeks who laid the groundwork for algebra. Imagine a time when numbers weren't just symbols, but represented real-world objects being counted and measured—how cool is that? Second, understanding how to factor can be quite beneficial in the real world! For example, in engineering and physics, many calculations involve polynomial expressions. Simplifying these through factoring can save time and lead to clearer understanding—whether it's designing bridges or optimizing manufacturing processes, polynomials are everywhere!