The polynomial function \( h \) is defined by \( h(x)=4 x^{3}+3 x^{2}-9 x-2 \)

Did you know that polynomial functions like \( h(x) = 4x^3 + 3x^2 - 9x - 2 \) can be visualized in beautiful ways? When you graph this function, you'll see it oscillating with curves that can have one, two, or even three turning points, depending on its critical points. The fascinating part is that the highest power of \( x \), in this case, three, indicates that it can have up to three real roots, which are the x-intercepts! In real-world applications, cubic polynomials appear in various fields such as physics, engineering, and economics. For instance, they can model the trajectory of objects in motion, where the cube term captures the acceleration due to gravity. You might encounter such functions when determining profit maximization or analyzing the volume of irregular shapes, showing just how versatile and practical polynomials can be!