how does the method of partial fraction decomposition by long division works when there is repeated root?

Question

I am reading A treatise on integral calculus and I can't understand what does the author means here, how does $\frac{y^2+2y+1}{y+2}$ turned into $0.5 +0.75y+0.125y^2-\frac{y^3}{8(2+y)}$

and I didn't understand the whole method

Answer 1

The original fraction is given by the following:

$$ \frac{\left(y+1\right)^{2}}{y^{3}\cdot(y+2)} $$

This is a ratio between a quadratic polynomial (which is simply a cubic polynomial with first coefficient zero) and a polynomial of order $4$. There are many ways to express a cubic polynomial, two examples are given below:

$$ \begin{aligned} p\left(y\right) &= ay^{3}+by^{2}+cy+d \\ p\left(y\right) &= A\left(y+2\right)+By\left(y+2\right)+Cy^{2}\left(y+2\right)+Dy^{3} \end{aligned} $$

Using both expression, our numerator is given by the followings, respectively:

$$ \begin{aligned} \left(y+1\right)^{2} &= y^{2}+2y+1 \\ \left(y+1\right)^{2} &=\frac{y+2}{2}+\frac{3y\left(y+2\right)}{4}+\frac{y^{2}\left(y+2\right)}{8}-\frac{y^{3}}{8} \end{aligned} $$

Substitute back to the original equation to obtain the desired result.

Key Points

• There are more than one way to express a polynomial
• Choose numerator expression that can cancel terms in the denominator, in our case $y^{3}$ and $y+2$