# Question: Fastest way to solve this? Can this be written as a matrix and solved via elimination? A, B, C, D and E are constants. x is arbitrary.

I would like to know if anyone has an idea what the best way to find the constants A, B, C, D, and E is. Can this be solved/written using a Matrix?

x can be chosen arbitrarily to find the constants A, B, C, D and E.

x4 - 5x3 - 30x2 - 36x = A*(x+1)2(x-2)(x+2)+B(x+1)(x-2)(x+2)+C*(x-2)(x+2)+D(x+1)3(x+2)+E(x+1)3*(x-2)

I know that the most efficient way is to chose x as -1, 2 and -2, so that the maximum amount of summands equal zero. Can this be solved faster, if we somehow write this as a Matrix and apply gaussian elimination?

This is part of solving an integral via partial fractions.

• I definitely wouldn't call it faster, but you can use a matrix. Just expand the right hand side, collect like terms of $x$, and equate coefficients. Just by looking at the $x^4$ terms, I can tell, for example, that it will begin with $(A + D + E)x^4 + \ldots$, and so the equation from the $x^4$ terms will be $A + D + E = 1$. The other powers will give you four more equations (don't forget the constant term), which you can solve simultaneously. This is by far the longer, more tedious route, but it can be done. Dec 11, 2022 at 16:51
• Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to. Dec 11, 2022 at 17:01
• No. The method of selecting $x = -1, 2, -2$ is the fastest method. Dec 12, 2022 at 2:31