Factor the denominator and then the
fraction may be written in the form:
(2x2 – x + 4)/(x3
+ 4x) = (2x2 – x + 4)/ [x(x2 + 4)] = A/x +
(Bx + C)/(x2 + 4)
Multiply both sides by x3 +
4x. Then for all x we have 2x2 – x + 4 = A(x2
+ 4) + (Bx + C)x .
Substitute x = 0 into the previous
identity. Then 4 = 4A and A = 1.
Write the coefficients at x and at x2: –1 = C, 2 = A + B, and B = 2 – A = 1.
Answer: (2x2 – x + 4)/(x3
+ 4x) =1/x + (x – 1)/(x2 + 4).
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