Factor the greatest common facto \( 4 x^{3}-4 x^{2}+20 x= \)

To factor the greatest common factor (GCF) from the expression \(4x^{3} - 4x^{2} + 20x\), we first identify the GCF of the coefficients 4, -4, and 20, which is 4. Additionally, each term includes at least one factor of \(x\), so we can also factor out \(x\). This gives us: \[ 4x(x^{2} - x + 5) \] Now, the expression is factored as \(4x(x^{2} - x + 5)\). Let's dive into some historical tidbits: Did you know that factoring polynomials dates back to ancient mathematicians like the Babylonians and Greeks? Their work laid the groundwork for modern algebra, but it wasn't until the Middle Ages that algebra began to resemble the form we use today, thanks to scholars from the Arabic world and Europe rejuvenating the study. Now, here’s a practical insight: In real-world applications, factoring can be incredibly useful! For instance, engineers often model systems or structures using polynomial equations. By factoring these equations, they can determine critical points like where loads might be maximized or minimized, ensuring that designs are both safe and effective!