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# find the partial fraction decomposition of (x+3)/x^2*(x^2+4)?

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- 2 months ago
a/x + b/x^2 + cx/(x^2 + 4) + d/(x^2 + 4) = (x + 3) / (x^2 * (x^2 + 4))

a * x * (x^2 + 4) + b * (x^2 + 4) + (cx + d) * x^2 = x + 3

ax^3 + 4ax + bx^2 + 4b + cx^3 + dx^2 = 0x^3 + 0x^2 + x + 3

ax^3 + cx^3 = 0x^3

a + c = 0

a = -c

bx^2 + dx^2 = 0x^2

b + d = 0

b = -d

4ax = x

4a = 1

a = 1/4

c = -1/4

4b = 3

b = 3/4

d = -3/4

(1/4) / x + (3/4) / x^2 - (1/4) * (x + 3) / (x^2 + 4) =>

(1/4) * (1/x + 3/x^2 - (x + 3) / (x^2 + 4))

Test

((1 * x * (x^2 + 4) + 3 * (x^2 + 4) - (x + 3) * x^2)) / (x^2 * (x^2 + 4)) =>

(x^3 + 4x + 3x^2 + 12 - x^3 - 3x^2) / (x^2 * (x^2 + 4)) =>

(4x + 12) / (x^2 * (x^2 + 4))

(1/4) * (4x + 12) / (x^2 * (x^2 + 4)) =>

(x + 3) / (x^2 * (x^2 + 4))

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