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 tums around at each zero \( f(x)=5(x-6)(x-1)^{2} \) Determine the zero(s) The zero(s) is/are 1.6 (Type integers or decimals. Use a comma to separate answers as needed) Determine the multiplicities of the zero(s). 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 largest zero is The multiplicity of the other zero is (Simplify your answers ) B. There are two zeros The multipliplicitity of the the smallest zero is (Simplify your answers ) C. There is one zero The multiplicity of the zero is 2 . The multiplicity of the smallest zero is 112 points (Simplity your answer )

To find the zeros of the polynomial function \( f(x) = 5(x-6)(x-1)^{2} \), we set each factor equal to zero. The first factor yields the zero \( x = 6 \), while the second factor gives \( x = 1 \). Thus, the zeros are \( 1 \) and \( 6 \). Next, we analyze the multiplicities. The factor \( (x - 1)^2 \) indicates that \( 1 \) has a multiplicity of \( 2 \) (it is repeated), while the factor \( (x - 6) \) indicates that \( 6 \) has a multiplicity of \( 1 \). Therefore, there are two distinct zeros: \( 1 \) (multiplicity 2) and \( 6 \) (multiplicity 1). At the zero \( x = 1 \), since the multiplicity is even, the graph touches the \( x \)-axis and turns around. At the zero \( x = 6 \), where the multiplicity is odd, the graph crosses the \( x \)-axis. 0, 6 (Type integers or decimals. Use a comma to separate answers as needed) There are two zeros. The multiplicity of the smallest zero is 2. The multiplicity of the largest zero is 1. When you're grappling with polynomial functions like this, it’s crucial to understand that the multiplicity of a zero influences the graph's behavior significantly. Even multiplicities create a "bounce" off the x-axis, while odd multiplicities allow the graph to pass through, quite literally crossing the line! Remember to sketch a quick graph as these properties can help visualize the function's behavior. Taking a closer look at polynomials and their properties can also be magically fascinating! You might want to dive into books or resources on polynomial functions, factoring techniques, and even calculus topics around derivatives and their relationships to polynomial graphs. It’s a journey, and each step can open up more exciting mathematical mysteries!