Factor. Check by multiplying. \( 4 x^{9} y^{7}+12 x^{7} y^{6}+32 x y \)

To factor \( 4 x^{9} y^{7}+12 x^{7} y^{6}+32 x y \), we start by identifying the greatest common factor (GCF) in all the terms. The GCF here is \( 4xy \). Factoring that out gives us: \[ 4xy (x^{8}y^{6} + 3x^{6}y^{5} + 8) \] Now, let's check by multiplying back: 1. Distributing \( 4xy \) to each term results in: - \( 4xy \cdot x^{8}y^{6} = 4x^{9}y^{7} \) - \( 4xy \cdot 3x^{6}y^{5} = 12x^{7}y^{6} \) - \( 4xy \cdot 8 = 32xy \) So, the original expression is confirmed as \( 4xy (x^{8}y^{6} + 3x^{6}y^{5} + 8) \). To add a sprinkle of historical background, did you know that factoring polynomials dates back all the way to ancient civilizations? The Babylonians used methods to solve quadratic equations, and the Greeks looked into geometric representations of polynomials. It’s amazing how these ancient techniques laid the foundation for modern algebra! For some real-world applications, factoring is immensely useful in optimization problems. Engineers often need to break down complex equations to find maximum efficiencies or minimum costs in designs - think bridge construction or resource allocation! Understanding how to factor polynomials helps simplify these analyses and make solutions more accessible.