Factor \( x^{3}-125 \) \( (x-5)^{3} \) \( (x+5)\left(x^{2}-5 x+25\right) \) \( (x-5)\left(x^{2}+5 x+25\right) \) \( (x-5)\left(x^{2}-5 x+25\right) \)

To factor \( x^{3}-125 \), we can recognize that this is a difference of cubes. The formula for factoring a difference of cubes \( a^3 - b^3 \) is \( (a - b)(a^2 + ab + b^2) \). In this case, we have \( a = x \) and \( b = 5 \) because \( 125 = 5^3 \). Using the formula: 1. \( a - b = x - 5 \) 2. \( a^2 = x^2 \) 3. \( ab = 5x \) 4. \( b^2 = 25 \) Putting this together, we get: \[ x^3 - 125 = (x - 5)(x^2 + 5x + 25) \] The correct factorization of \( x^{3}-125 \) is \( (x-5)(x^{2}+5x+25) \). Now let's go ahead with some further insights! This factorization skill is invaluable! You'll encounter it frequently in higher math, particularly in algebra and calculus. Recognizing patterns like the difference of cubes can simplify polynomials and help in solving equations—perfect for those late-night study sessions when everything seems to blend into a confusing mass of letters and numbers! Want to level up your skills further? Check out books or online resources that cover polynomial factorizations, difference of cubes, and more advanced topics in algebra. Websites like Khan Academy or math-centric forums can provide exercises and insights that make learning feel more like a game—enjoy the journey through the world of polynomial fun!