Alternative efficient ways to do partial fraction?

[hanhao]

what are alternative ways to do partial fraction?
other than the normal method of mulplying the whole term over and comparing coefficient which is super headache method

i heard of "cover-up method", gets the answer real quick

anyone have details to this?

 

[TD]

By choosing "smart zeroes" of the denominator, you can speed up the process but this won't always work (fully).

For example, we wish to expand:

$$\frac{{x + 1}} {{x^3 - 7x + 3}}$$

So you start by factoring and introducing the unknown coëfficiënts:

$$\frac{{x + 1}} {{x^3 - 7x + 3}} = \frac{{x + 1}} {{\left( {x - 1} \right)\left( {x - 2} \right)\left( {x + 3} \right)}} = \frac{A} {{x - 1}} + \frac{B} {{x - 2}} + \frac{C} {{x + 3}} = \frac{{A\left( {x - 2} \right)\left( {x + 3} \right) + B\left( {x - 1} \right)\left( {x + 3} \right) + C\left( {x - 1} \right)\left( {x - 2} \right)}} {{\left( {x - 1} \right)\left( {x - 2} \right)\left( {x + 3} \right)}}$$

Then you set up the equation for the numerators to be the same:

$$A\left( {x - 2} \right)\left( {x + 3} \right) + B\left( {x - 1} \right)\left( {x + 3} \right) + C\left( {x - 1} \right)\left( {x - 2} \right) = x + 1$$

Traditionally, you would now expand the entire left side and then group in powers of x to get a system, which is still doable.

Another way to go on now would be to choose some values for x, preferably values which may simplify things, i.e. zeroes of the denominator:

$$\begin{gathered} x = 1 \Rightarrow A\left( {1 - 2} \right)\left( {1 + 3} \right) = 2 \Leftrightarrow A = - \frac{1} {2} \hfill \\ x = 2 \Rightarrow B\left( {2 - 1} \right)\left( {2 + 3} \right) = 3 \Leftrightarrow B = \frac{3} {5} \hfill \\ x = - 3 \Rightarrow C\left( { - 3 - 1} \right)\left( { - 3 - 2} \right) = - 2 \Leftrightarrow C = - \frac{1} {{10}} \hfill \\ \end{gathered}$$

Which gives the result:

$$\frac{{x + 1}} {{x^3 - 7x + 3}} = \frac{{ - 1}} {{2\left( {x - 1} \right)}} + \frac{3} {{5\left( {x - 2} \right)}} - \frac{1} {{10\left( {x + 3} \right)}}$$

 

[quicknote]

Thank you for your question. The "cover-up method" is a commonly used shortcut in partial fraction decomposition that can make the process quicker and easier. It involves covering up the variable in the denominator with a value that will make the fraction equal to zero, and then solving for the unknown coefficient. This method works well for simple fractions with linear denominators, but may not be as effective for more complex fractions. Another alternative method is the Heaviside cover-up method, which involves using a formula to determine the coefficients without having to solve for them individually. This method is useful for fractions with repeated linear factors. Additionally, there are computer programs and calculators that can perform partial fraction decomposition for more complex fractions. These methods can save time and reduce the chances of making errors, but it is important to understand the underlying concepts of partial fraction decomposition in order to use these shortcuts effectively.