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{-3y^{3}+6y}{y^{3}(y + 2)^{2}(y - 3)(y + 3)^{2}}=\frac{a}{(?)}+\frac{b}{(?)}+\frac{c}{(?)}+cdots) ((y)) ((y - 3)) ((y)^{2}) ((y - 3)^{2}) ((y)^{3}) ((y - 3)^{3}) ((y + 2)) ((y + 3)) ((y + 2)^{2}) ((y + 3)^{2}) ((y + 2)^{3}) ((y + 3)^{3})

Explanation:

Step1: Recall partial - fraction rules

For a rational function $\frac{f(y)}{g(y)}$ where $g(y)=y^{3}(y + 2)^{2}(y - 3)(y + 3)^{2}$, in partial - fraction decomposition, for a linear factor $ay + b$ with multiplicity $n$, the terms in the decomposition are of the form $\frac{A_1}{ay + b}+\frac{A_2}{(ay + b)^{2}}+\cdots+\frac{A_n}{(ay + b)^{n}}$.

Step2: Identify denominators for each factor

For the factor $y$ with multiplicity $3$, the denominators are $y$, $y^{2}$, $y^{3}$. For the factor $(y + 2)$ with multiplicity $2$, the denominators are $(y + 2)$ and $(y + 2)^{2}$. For the factor $(y - 3)$ with multiplicity $1$, the denominator is $(y - 3)$. For the factor $(y + 3)$ with multiplicity $2$, the denominators are $(y + 3)$ and $(y + 3)^{2}$.

Answer:

$(y)$, $(y)^{2}$, $(y)^{3}$, $(y + 2)$, $(y + 2)^{2}$, $(y - 3)$, $(y + 3)$, $(y + 3)^{2}$