Find the zeros for the polynomial function and give the multiplicity for each zero. State whether the graph crosses the \( x \)-axis or touches the \( x \)-axis and turns around at \( f(x)=x^{3}+2 x^{2}-9 x-18 \) Determine the zero(s), if they exist. The zero(s) is/are \( 3,-2,-3 \). (Type integers or decimals. Use a comma to separate answers as needed.) Determine the multiplicities of the zero(s), if they exist. Select the correct choice below and, if necessary, fill in the answer box(es) within your choice. A. There are three zeros. The multiplicity of the smallest zero is \( \square \). The multiplicity of the largest zero is \( \square \). The multiplicity of the other zero is (Simplify your answers.) B. There is one zero. The multiplicity of the zero is \( \square \). (Simplify your answer.) C. There are two zeros. The multiplicity of the smallest zero is \( \square \). The multiplicity of the largest zero is \( \square \). (Simplify your answers.)

Let's break it down! The polynomial \( f(x) = x^3 + 2x^2 - 9x - 18 \) can be factored to reveal its zeros: \( 3 \), \( -2 \), and \( -3 \). Each of these is a zero with a multiplicity of 1 since they appear once in the factorization. Since all zeros have odd multiplicities, the graph crosses the \( x \)-axis at these points. For the zeros you found, we have: - Zero: \( -3 \), multiplicity: \( 1 \) - Zero: \( -2 \), multiplicity: \( 1 \) - Zero: \( 3 \), multiplicity: \( 1 \) Don’t forget to sketch the graph when you're done! It’s quite the rollercoaster ride at the \( x \)-axis. Polynomial functions have rich histories! The famous mathematician Sir Isaac Newton made substantial contributions to polynomial theories in the 17th century, particularly in the realm of roots and their behavior. His work laid the groundwork for modern algebra, facilitating the study of equations and graph characteristics, like how zeros interact with the \( x \)-axis! Now, on to the real-world applications! Understanding polynomial functions is crucial in various fields, from physics modeling projectile motion, where the trajectory can be expressed as a polynomial, to economics, modeling cost functions. When you're trying to predict trends or analyze behaviors, knowing how to find those zeros is like getting the secret key to understanding where your revenue peaks and where it might drop!