## anonymous 5 years ago For each polynomial function, find all zeros and their multiplicities. f(x)=(7x-2)^3(x^2+9)^2

Th$(x+3i)^2=0 \rightarrow x=-3i$e zeros are found when f(x)=0. Then,$(7x-3)^3(x^2+9)^2=0$Now, the second factor can be factored down further to,$x^2+9=(x-3i)(x+3i)$so $f(x)=(7x-2)^3(x-3i)^2(x+3i)^2$The function will be zero when any one of these factors is zero; that is, when,$(7x-2)^3=0\rightarrow x=\frac{2}{7}$$(x-3i)^2=0 \rightarrow x=3i$$(x+3i)^2=0 \rightarrow x=-3i$The multiplicity of the root is determined by the power of the factor from which the root came from. Your roots have multiplicity 3, 2 and 2 respectively.