What are the zeros of the polynomial function \( f(x) = x^{3} - 6x^{2} + 9x \)?

To find the zeros of the polynomial function \( f(x) = x^{3} - 6x^{2} + 9x \), we start by factoring the polynomial. First, we can factor out an \( x \): \[ f(x) = x(x^2 - 6x + 9) \] Next, we notice that the quadratic \( x^2 - 6x + 9 \) can be factored further since it is a perfect square: \[ x^2 - 6x + 9 = (x - 3)^2 \] Putting it all together, we have: \[ f(x) = x(x - 3)^2 \] Setting \( f(x) = 0 \) gives us: \[ x(x - 3)^2 = 0 \] This gives us the zeros: 1. \( x = 0 \) 2. \( x - 3 = 0 \) or \( x = 3 \) (with a multiplicity of 2) So, the zeros of the polynomial function are \( x = 0 \) and \( x = 3 \). To add a bit of historical spice, polynomials date back to ancient civilizations, such as the Babylonians and Greeks, who solved cubic equations using geometric methods. It's fascinating how the foundations they laid down thousands of years ago led to the algebraic explorations we enjoy today! For real-world applications, understanding polynomial functions and their zeros is crucial in fields like engineering, physics, and economics. Engineers use polynomial equations to model systems and optimize designs, while economists rely on them to forecast trends and make financial predictions. It’s a powerful tool that connects the abstract world of math to practical, real-life scenarios!