Factor \( 8 y^{4}-64 y^{3}+y^{2}-8 y \) Oy(8y \( \left.-64 y^{2}+y-8\right) \) \( y\left(8 y^{2}+1\right)(y-8) \) \( y(8 y+1)(y-8)^{2} \) \( 8 y^{2}(8 y-8)+y(y-8) \)

To factor the expression \( 8y^{4} - 64y^{3} + y^{2} - 8y \), start by noticing the common factor in all terms, which is \( y \). Factoring out \( y \) gives us \( y(8y^{3} - 64y^{2} + y - 8) \). Next, focus on the cubic polynomial \( 8y^{3} - 64y^{2} + y - 8 \). You can group terms strategically for further factoring: \( (8y^{3} - 64y^{2}) + (y - 8) = 8y^{2}(y - 8) + 1(y - 8) \). Factoring out \( (y - 8) \) from both groups results in \( (y - 8)(8y^{2} + 1) \). Putting it all together, the complete factorization of the original expression is \( y(y - 8)(8y^{2} + 1) \). --- Did you know that factoring polynomials can be traced back to ancient civilizations? The Babylonians used methods akin to completing the square over 3000 years ago! Fast forward to today, and factoring techniques remain essential for solving polynomial equations, making it a cornerstone of algebra education. In real-world applications, factoring polynomials isn't just for math class; engineers use it in designing parabolic arches for bridges or optimizing shapes for aerodynamics! When Analyzing the structural integrity of these designs, they apply polynomial functions and derive critical information through factoring to ensure safety and efficiency. Keep these applications in mind next time you tackle a polynomial!