# systems of Nonlinear Equations Help..... points?

The partial fraction decomposition of:

x^2 + 2 / x^3 + x^2

can be written in the form of:

(f(x) / x) + (g(x) / x^2) + (h(x) / x + 1)

f(x)=_________

g(x)=_________

h(x)=_________

Relevance

First identify the denominators of the partial fractions

(x^2 + 2)/(x^3 + x^2) = (x^2 + 2)/[x^2.(x + 1)]

which indicates that possible denominators include x, x^2 and (x + 1). If we select all of these, it should be possible to decompose the LH expression into the form

(x^2 + 2)/(x^3 + x^2) = A/x + B/x^2 + C/(x + 1)

where A, B and C are numbers. If we recombine the partial fractions on the right, we obtain for the numerators

(x^2 + 2) = A.x.(x + 1) + B.(x + 1) + C.x^2

Substituting x = -1 into this equation gives C = 3

Substituting x = 0 similarly gives B = 2

Because the coefficient of x on the LHS is zero, A + B = 0, and hence A = -2

Hence the initial expression may be decomposed into

(x^2 + 2)/(x^3 + x^2) = -2/x + 2/x^2 + 3/(x + 1)

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