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| Math Central | Quandaries & Queries |
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Question from taiwo, a student: i copied the question below from john bird's higher engineering mathematics x^3 + 4^2 + 20x - 7 over (x-1)^2 (x^+8). i will appreciate it if you help me out so i can |
Hi Tiawo,
I think you mean x3 + 4x2 + 20 x - 7 over (x - 1)2(x2 + 8). The task is to write this in partial fraction form as
Writing this sum of three fractions with a common denominator give a numerator which must equal the numerator of the original fraction. Thus
x3 + 4x2 + 20 x - 7 = A(x - 1)(x2 + 8) + B(x2 + 8) + (Cx + D)(x - 1)2
Substituting x = 1 in this equation gives B = 2 and hence the equation simplifies to
x3 + 2x2 + 20 x - 23
= A(x - 1)(x2 + 8) + (Cx + D)(x - 1)2
= (A + C)x3 + (A - 2C + D)x2 + (8a + C - 2D)x + (-8A + D)
Equating polynomial coefficients leads to the linear system
A + C = 1
-A - 2C + D = 2
8A + C - 2D = 20
-8A + D = -23
This system yields A = 3, C = -2 and D = 1.
I hope this helps,
Harley
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