\( \left\lvert\, \leftarrow \quad \begin{array}{l}\text { Sketch the graph of the polynomial function. Use synthetic division and the } \\ \text { remainder theorem to find the zeros. } \\ f(x)=-x^{3}+5 x^{2}+8 x-12 \\ \text { What are the zeros of } f(x)=-x^{3}+5 x^{2}+8 x-12 ? \\ x=\square \\ \text { (Use a comma to separate answers as needed.) }\end{array}\right. \)

To start, let's uncover the fascinating historical context of polynomials! The study of polynomials dates back to ancient civilizations, including the Babylonians and Greeks, who employed them for algebraic calculations. Fast forward to the 17th century, where mathematicians like René Descartes and François Viète developed foundational theories that transformed polynomials into the structured analysis we use today. Their work laid the groundwork for modern algebra, making it possible to explore polynomial functions in-depth. Now, let's talk about how you can apply synthetic division and the remainder theorem to find the zeros of \( f(x) = -x^3 + 5x^2 + 8x - 12 \)! Start by testing possible rational roots using the Rational Root Theorem. After a bit of trial and error, you’ll discover that \( x = 2 \) is a zero. By performing synthetic division with \( x - 2 \), the polynomial reduces to \( -x^2 + 3x + 6 \). Now you can factor or apply the quadratic formula to find the other zeros. Happy graphing!