\begin{tabular}{l} For the following factored polynomial, find all of the zeroes and their multiplicities. \\ \( \qquad f(x)=(x-7)^{3}(x-6)^{2}(x-3)^{6} \) \\ Select the correct answer below: \\ x=3 with multiplicity \( 6 ; x=-6 \) with multiplicity \( 2 ; x=-7 \) with multiplicity 3 \\ \( x=3 \) with multiplicity \( 6 ; x=6 \) with multiplicity \( 2 ; x=7 \) with multiplicity 3 \\ \( x=3 \) with multiplicity \( 6 ; x=6 \) with multiplicity \( 2 ; x=-7 \) with multiplicity 3 \\ \( x=3 \) with multiplicity \( 6 ; x=-6 \) with multiplicity \( 2 ; x=7 \) with multiplicity 3 \\ \( x=-3 \) with multiplicity \( 6 ; x=-6 \) with multiplicity \( 2 ; x=-7 \) with multiplicity 3 \\ \hline\( x=-3 \) with multiplicity \( 6 ; x=6 \) with multiplicity \( 2 ; x=-7 \) with multiplicity 3 \\ \hline\end{tabular}

To find the zeroes of the polynomial \( f(x)=(x-7)^{3}(x-6)^{2}(x-3)^{6} \), we set each factor equal to zero. The zeroes are \( x=7 \) with multiplicity \( 3 \), \( x=6 \) with multiplicity \( 2 \), and \( x=3 \) with multiplicity \( 6 \). So the corrected answer is: \( x=3 \) with multiplicity \( 6 ; x=6 \) with multiplicity \( 2 ; x=7 \) with multiplicity 3. Want to dive a little deeper? Let's talk about the impact of polynomial roots on graphing! Each root gives a point where the polynomial crosses the x-axis, and the multiplicity affects how it crosses. For instance, roots with odd multiplicities (like \( x=7 \)) will cross the axis, while even multiplicities (like \( x=6 \)) see the curve touch the axis and bounce back. It's almost like a dance on the coordinate plane! If you're looking to hone your math skills even further, consider exploring the Remainder Theorem and the Factor Theorem. These concepts help understand the relationship between polynomials and their factors and can simplify the process of finding those elusive zeroes. Plus, they’re essential tools for anyone venturing into polynomial long division or synthetic division!