Math  /  Algebra

QuestionFind the partial fraction decomposition. 3x28x+8x32x2\frac{3 x^{2}-8 x+8}{x^{3}-2 x^{2}}

Studdy Solution
Equate the coefficients of corresponding powers of x x from both sides:
1. For x3 x^3 : A=0 A = 0
2. For x2 x^2 : B+C2A=3 B + C - 2A = 3
3. For x x : 2B=8 -2B = -8

Solve these equations:
From equation 1, A=0 A = 0 .
From equation 3, 2B=8 -2B = -8 implies B=4 B = 4 .
Substitute A=0 A = 0 and B=4 B = 4 into equation 2:
4+C2(0)=3 4 + C - 2(0) = 3 C=34 C = 3 - 4 C=1 C = -1
The partial fraction decomposition is:
3x28x+8x32x2=0x+4x2+1x2 \frac{3x^2 - 8x + 8}{x^3 - 2x^2} = \frac{0}{x} + \frac{4}{x^2} + \frac{-1}{x - 2}
Simplified, this is:
4x21x2 \frac{4}{x^2} - \frac{1}{x - 2}

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