What is the factored form of the polynomial? \( x^{2}-16 x+48 \) \( (x-4)(x-12) \) \( (x-6)(x-8) \) \( (x+4)(x+12) \) \( (x+6)(x+8) \)

To factor the polynomial \( x^{2} - 16x + 48 \), we can look for two numbers that multiply to 48 (the constant term) and add up to -16 (the coefficient of the linear term). The numbers -12 and -4 fit this requirement, which means the polynomial can be factored as \( (x - 12)(x - 4) \). So the correct factored form is \( (x - 4)(x - 12) \). Now let’s break it down: Imagine \( x \) represents passing time, with 16 x being the total time taken, and 48 being 48 tasks at hand. If you split these tasks perfectly into two groups that each take 4 and 12 hours, you’re managing your time effectively! For more insight, this polynomial is a classic quadratic expression! In high school algebra, students often encounter it when learning about factoring techniques like grouping or the diamond method. It’s a fun puzzle akin to finding a combination lock where balance in input sets the path to problem-solving.