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Expert: - 10/3/2008


Question
QUESTION: Express the following in partial fraction: (x^2+23)/((x+1)^3(x-2))


i have tried it many times for hours. Much appreciated.

ANSWER: The equations are A/(x+1)^3 + B/(x+1)² + C/(x+1) + D/(x-2).
This can be all put over (x+1)^3(x-2) by multiplying each term by what's missing.  The result is

[A(x-2) + B(x+1)(x-2) + C(x+1)²(x-2) + D(x+1)^3]/[(x+1)^3(x-2)].

Multiplying out each of the terms gives
A(x-2)=A(x-2), B(x+1)(x-2)=B(x²-x-2), C(x+1)²(x-2)=C(x^3-3x-2), and
D(x+1)^3=D(x^3+3x²+3x+1).

Putting these equations back in and grouping like terms gives us
(C+D)x^3 + (B+3D)x² + (A-B-3C+3D)x + -2A-2B-2C+D

From x²+23, we know that C+D=0, B+3D=1, A-B-3C+D=0, and -2A-2B-2C+D=23.

This comes down to solving the following matrix:
 0   0   1 1 0
 0   1   0 3 1
 1 –1 –3 1 0
-2 –2  –2 1 0,
where each of the columns are A, B, C, D, and the solution, respectively.

Solving this and putting back in A, B, C, and D gave me the right answer.  I stuck with the element that was one each time and zeroed out the others up until the last time.  On the last time, I got a lot of numbers in 23rd s.


---------- FOLLOW-UP ----------

QUESTION: I dont know how to use matrix to solve..Any other way you know how to solve it..I have get the 4 equations already i have try to solve it using substitution & elimination i still got wrong answers..The real answers for it is 1/(x-2) - 1/(x+1) - 2/(x+1)^2 -8 /(x+1)^3


much obliged.

Answer
To solve a matrix, we need a 1 in each column and the rest of the values 0.  When we're done, there will be a 1 in each row and the rest of the elements will be 0 if there is a solution.

Using the matrix
0  0  1 1 0
0  1  0 3 1
1 –1 –3 1 0
-2 –2 –2 1 0, where the first row is for x1, the second for x2, the third for x3, the fourth for x4, and the last column is for the solutions.

The first, second and third rows would be kept the same.  The fourth row needs to have 2*3rd row added to it, giving a new matrix

0  0  1 1 0
0  1  0 3 1
1 –1 –3 1 0
0 -4 -8 3 0.

In this case, keep the first and second rows the same.  To the third row, add the second row.  To the fourth row, add 4 * the second row.
The result is

0 0  1  1 0
0 1  0  3 1
1 0 -3  4 1
0 0 -8 15 4.

Divide the last row by -8, add 3/8 times the last row to the third row, and add 1/8 to the first row.

After this is done, you'll have a few fractions involved.  Use row 1 as the final row and subtract it from the rest.

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