Express 1/x⁴ + 5x² + 6 into partial fraction

Question

1/x⁴ + 5x² + 6

Express in partial fraction

Answer 1

I assume that the denominator is PARENTHESIZED and consists of (x^2 + 3)(x^2 + 2).

 

Therefore this can be represented as (Ax+B)/(x^2+2) + (Cx+D)/(x^2+3)

 

When we obtain a common denominator AND THEN EQUATE THE NUMERATORS we get

1 = (Ax+B)(x^2+3) + (Cx+D)(x^2+2). 

 

Then use of FOIL obtains

1 = Ax^3 + 3Ax + Bx^2 + 3B   +   Cx^3 +2Cx + Dx^2 + 2D.

 

When we equate coefficients of the various powers of x on the two sides of this equation we get the following system:

A+C = 0

B+D = 0

3A+2C = 0

3B+2D = 1.

 

This I would solve by using the rref capability of my old TI-83 graphing calculator, if I had not given it away. You can use many different methods of solving this 4x4 system of linear equations, including Cramer's rule or Gauss-Jordan method.