Question

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

Answer 1

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.