Write out the partial fraction decomposition of the following rational expression.
3x^2-7x-2/x^3-x
Multiply all terms by x^3.
3x^5….
To find the partial fraction decomposition of the rational expression (3x^2 - 7x - 2) / (x^3 - x), we need to factor the denominator first.
We start by factoring out the greatest common factor (GCF) from the denominator:
x(x^2 - 1)
Next, we can factor the (x^2 - 1) term further using the difference of squares formula:
x(x - 1)(x + 1)
Now, we have factored the denominator completely as x(x - 1)(x + 1).
To find the partial fraction decomposition, we assume that the rational expression can be written as a sum of two or more simpler fractions with individual denominators equal to the factors of the original denominator.
Therefore, we can write:
(3x^2 - 7x - 2) / (x^3 - x) = A / x + B / (x - 1) + C / (x + 1)
Our goal now is to find the values of A, B, and C.
To find the value of A, we multiply the entire equation by the denominator x to eliminate the fraction:
(3x^2 - 7x - 2) = A(x - 1)(x + 1) + Bx(x + 1) + Cx(x - 1)
Expanding the equation:
3x^2 - 7x - 2 = A(x^2 - 1) + Bx(x + 1) + Cx(x - 1)
Simplifying further:
3x^2 - 7x - 2 = A(x^2 - 1) + B(x^2 + x) + C(x^2 - x)
Now, we can equate the coefficients of like terms on both sides to find the values of A, B, and C.
For x^2 term:
0 = A + B + C
For x term:
-7 = B - C
For constant term:
-2 = -A
From the last equation, we find that A = 2.
Substituting A = 2 into the first equation, we have:
0 = 2 + B + C
-2 = B + C
From the second equation, we get B + C = -2. By subtracting this from the third equation B - C = -7, we can solve for B and C.
(B + C) - (B - C) = -2 - (-7)
2C = 5
C = 5/2
Substituting C = 5/2 back into B + C = -2, we can solve for B:
B + 5/2 = -2
B = -9/2
Therefore, we have found the values of A, B, and C to be A = 2, B = -9/2, and C = 5/2.
So, the partial fraction decomposition of the given rational expression is:
(3x^2 - 7x - 2) / (x^3 - x) = 2 / x - (9/2) / (x - 1) + (5/2) / (x + 1)