Application: Factoring Grading Focus: Empirical \& Quantitative Skills, Critical Thinking Skills Factor the following polynomial: \( x^{2}-8 x+15 \) (A) \( (x+3)(x+5) \) (B)

To factor the polynomial \( x^{2}-8x+15 \), we start by looking for two numbers that multiply to \( 15 \) (the constant term) and add up to \( -8 \) (the coefficient of \( x \)). The numbers \( -3 \) and \( -5 \) work perfectly since \( -3 \times -5 = 15 \) and \( -3 + (-5) = -8 \). Therefore, the factored form is \( (x-3)(x-5) \). Now, if you checked the answer against the original polynomial by expanding it, you would find that \( (x-3)(x-5) = x^2 - 5x - 3x + 15 = x^2 - 8x + 15 \), confirming that your factoring was spot on! Now, let’s talk about practical applications! Factoring is not just an academic exercise but a vital skill in fields like engineering and physics. For example, when dealing with quadratic equations that model real-world situations—like projectile motion—the ability to factor these equations can help predict the behavior of objects over time, such as calculating the optimal angle for launching a rocket. Also, a common mistake when factoring is forgetting to check if there are any common factors first. For instance, if you encountered a polynomial like \( 2x^2 - 16x + 30 \), it’s easy to jump straight to factoring without first factoring out the GCF of 2. Always remember to simplify first; it makes the process much easier!