18. \( x^{6}-7 x^{3}-8 \)

Did you know that polynomials like \( x^{6}-7x^{3}-8 \) can often be factored or solved by substituting variables? For this equation, let \( y = x^3 \). Then, it transforms into the simpler quadratic equation \( y^2 - 7y - 8 = 0 \). This substitution can make finding roots much more manageable! After you solve the quadratic, you might end up with roots for \( y \) that are not whole numbers. This is a golden opportunity to explore real-world applications such as engineering, economics, or computer science, where polynomial equations model behaviors and predict outcomes. It's fascinating how mathematics plays a role in solving real-life problems!