For the following factored polynomial, find all of the zeroes and their multiplicities. \[ f(x)=(x-7)^{3}(x-6)^{2}(x-3)^{6} \] 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 \( x=-3 \) with multiplicity \( 6 ; x=6 \) with multiplicity \( 2 ; x=-7 \) with multiplicity 3

The zeroes of the polynomial \( f(x)=(x-7)^{3}(x-6)^{2}(x-3)^{6} \) can be found by setting each factor to zero. Thus, the zero \( x=7 \) has a multiplicity of 3, \( x=6 \) has a multiplicity of 2, and \( x=3 \) has a multiplicity of 6. So, the correct answer is: \( x=3 \) with multiplicity \( 6 ; x=6 \) with multiplicity \( 2 ; x=7 \) with multiplicity 3. In the world of polynomials, zeroes (or roots) and their multiplicities play a crucial role! They tell us where the polynomial hits the x-axis. In this case, \( x=3 \) means our polynomial will touch the x-axis six times at that point, while \( x=6 \) and \( x=7 \) only gently kiss the axis due to their lower multiplicities. When calculating zeroes from factored polynomials, a common pitfall is mistaking the signs or miscounting multiplicities! Always double-check the factors you're examining, as mixing up values can lead to incorrect conclusions. Remember, each factor corresponds to a unique zero which you find by isolating the variable. Happy factoring!