Partial fraction decomposition x-3/x^3+3x

Answer 1

To perform partial fraction decomposition, start by factoring the denominator of the fraction, x^3 + 3x.

x^3 + 3x = x(x^2 + 3)

Since there are no repeated factors in the denominator, the partial fraction decomposition will have the following form:

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

To find the values of A, B, and C, multiply both sides of the equation by x^3 + 3x:

x-3 = A(x^2 + 3) + Bx(x^2 + 3) + Cx^3

Expanding and collecting like terms:

x-3 = Ax^2 + 3A + Bx^3 + 3Bx + Cx^3

Now, equating the coefficients of each term:

1x^3 = Bx^3 + Cx^3
1 = B + C

0x^2 = Ax^2
0 = A

1x = 3Bx
1 = 3B
B = 1/3

Substituting B into the equation 1 = B + C:

1 = 1/3 + C
C = 2/3

Therefore, the partial fraction decomposition of x-3/(x^3+3x) is:

x-3/(x^3+3x) = 0/x + 1/(3x^2) + 2/(3(x^2 + 3))