Question

the local zoo closed its penguin exhibit, so puzzled penguin is trying to make a new living as a celebrity skin and body care product influencer. it barely pays rent, but he refuses to work for only penguins, so he has a side - gig as an online math tutor.
unfortunately, a penguins brain is the size of a large walnut. help puzzled penguin earn his daily fish by showing him how to perform partial fraction decomposition.
for the function shown below, select all denominators which would be included in the partial fraction decomposition:
\\(\frac{-3w^{3}+2w^{2}-5w - 1}{w^{3}(w - 3)^{3}(w - 1)^{3}(w + 2)^{2}}=\frac{a}{(?)}+\frac{b}{(?)}+\frac{c}{(?)}+\cdots\\)
\\(\square(w)\\) \\(\square(w - 1)\\)
\\(\square(w)^{2}\\) \\(\square(w - 1)^{2}\\)
\\(\square(w)^{3}\\) \\(\square(w - 1)^{3}\\)
\\(\square(w - 3)\\) \\(\square(w + 2)\\)
\\(\square(w - 3)^{2}\\) \\(\square(w + 2)^{2}\\)
\\(\square(w - 3)^{3}\\) \\(\square(w + 2)^{3}\\)

Explanation:

Step1: Recall partial - fraction decomposition rule

For a rational function $\frac{f(w)}{g(w)}$ where $g(w)=w^3(w - 3)^3(w - 1)^3(w + 2)^2$, and the general form of partial - fraction decomposition of $\frac{f(w)}{g(w)}$ has terms of the form $\frac{A_i}{w^i},\frac{B_i}{(w - 3)^i},\frac{C_i}{(w - 1)^i},\frac{D_i}{(w + 2)^i}$ with $i = 1,2,\cdots$ up to the power of the corresponding factor in the denominator of $g(w)$.

Step2: List all possible denominators

For the factor $w^3$, the denominators are $w$, $w^2$, $w^3$. For the factor $(w - 3)^3$, the denominators are $(w - 3)$, $(w - 3)^2$, $(w - 3)^3$. For the factor $(w - 1)^3$, the denominators are $(w - 1)$, $(w - 1)^2$, $(w - 1)^3$. For the factor $(w + 2)^2$, the denominators are $(w + 2)$, $(w + 2)^2$.

Answer:

$(w)$, $(w)^2$, $(w)^3$, $(w - 3)$, $(w - 3)^2$, $(w - 3)^3$, $(w - 1)$, $(w - 1)^2$, $(w - 1)^3$, $(w + 2)$, $(w + 2)^2$