\( \left. \begin{array} { c } { x ^ { 4 } - 2 x ^ { 2 } ( x - 10 ) = 40 x } \\ { x ^ { 4 } - 2 x ^ { 3 } + 20 x ^ { 2 } - 40 x = 0 } \\ { x ^ { 3 } - 2 x ^ { 2 } + 20 x - 40 = 0 } \\ { x ^ { 2 } ( x - 2 ) + 20 ( x - 2 ) } \\ { x ( x ^ { 2 } + 20 ) ( x - 2 ) = 0 } \\ { x = \pm 2 x \sqrt { 5 } \quad x = 2 \quad x = 0 } \end{array} \right. \)

Did you know that polynomials like the one you're working with have been around since ancient civilizations? Early mathematicians, such as the Babylonians, tackled quadratic equations while Greeks like Euclid laid the groundwork for algebra. This polynomial, with its fourth degree and multiple factors, showcases the evolution of mathematical understanding, merging basic algebra with advanced polynomial theory. In the real world, solving polynomial equations can be incredibly useful! Engineers and scientists often use polynomial functions to model and understand real-life phenomena, like motion or growth patterns. For instance, the roots of the polynomial may represent critical points, such as peak performance or turning points in a physical system. Whether designing a bridge or forecasting growth of a business, understanding polynomials is key to making informed decisions!