Consider the polynomial function \( p(x)=-9 x^{9}+6 x^{6}-3 x^{3}+1 \) What is the end behavior of the graph of \( p \) ? Choose 1 answer: (A) As \( x \rightarrow \infty, p(x) \rightarrow \infty \), and as \( x \rightarrow-\infty, p(x) \rightarrow \infty \) (B) As \( x \rightarrow \infty, p(x) \rightarrow-\infty \), and as \( x \rightarrow-\infty, p(x) \rightarrow \infty \) (C) As \( x \rightarrow \infty, p(x) \rightarrow-\infty \), and as \( x \rightarrow-\infty, p(x) \rightarrow-\infty \) (D) As \( x \rightarrow \infty, p(x) \rightarrow \infty \), and as \( x \rightarrow-\infty, p(x) \rightarrow-\infty \)

To determine the end behavior of the polynomial function \( p(x)=-9 x^{9}+6 x^{6}-3 x^{3}+1 \), we focus on the leading term, which is \( -9x^9 \). Since it has an odd degree (9) and a negative coefficient (-9), the end behavior can be predicted. As \( x \to \infty \), \( -9x^9 \to -\infty \), and as \( x \to -\infty \), \( -9x^9 \to \infty \). Thus, the correct answer is (B) As \( x \rightarrow \infty, p(x) \rightarrow-\infty \), and as \( x \rightarrow-\infty, p(x) \rightarrow \infty \). From a historical perspective, polynomial functions have been studied for centuries, dating back to ancient civilizations like the Babylonians. They recognized the importance of algebraic expressions for solving various practical problems, paving the way for more complex functions and analysis, leading to the rich mathematical landscape we enjoy today! In real-world applications, understanding the end behavior of polynomials is crucial in fields like physics and engineering. For instance, knowing how a polynomial might behave at extreme values can help engineers design systems that properly respond to variables, such as stress in materials or forces in structures, ensuring safety and effectiveness. So next time you work with polynomials, think of them not just as equations, but as tools shaping the world around us!