Express (x^6-x^5-4x^2+x)/(x^4+3x^2+2) in partial fraction. Help anybody? Much appreciated.
To express the given expression as partial fractions, follow these steps:
1. Factorize the denominator: x^4 + 3x^2 + 2. It can be factored as (x^2 + 1)(x^2 + 2).
2. Write the expression with unknown coefficients for the partial fractions. The numerator will have a linear term for each factor in the denominator.
(x^6 - x^5 - 4x^2 + x) / (x^4 + 3x^2 + 2) = A/(x^2 + 1) + B/(x^2 + 2).
3. Clear the denominator by multiplying both sides of the equation by (x^4 + 3x^2 + 2).
(x^6 - x^5 - 4x^2 + x) = A(x^2 + 2) + B(x^2 + 1).
4. Expand the equation.
x^6 - x^5 - 4x^2 + x = (A + B)x^2 + 2A + B.
5. Equate the coefficients of like powers of x on both sides of the equation.
For the coefficient of x^6: 0 = 0.
For the coefficient of x^5: -1 = 0.
For the coefficient of x^4: 0 = A + B.
For the coefficient of x^2: -4 = 2A + B.
For the constant term: 0 = 2A.
6. Solve the system of equations. Since A = 0 from the last equation, substitute A = 0 in the previous equation to get:
-4 = B.
Therefore, A = 0 and B = -4.
7. Substitute the values of A and B back into the equation:
(x^6 - x^5 - 4x^2 + x) / (x^4 + 3x^2 + 2) = 0/(x^2 + 1) + (-4)/(x^2 + 2).
Simplifying further, we get:
(x^6 - x^5 - 4x^2 + x) / (x^4 + 3x^2 + 2) = -4/(x^2 + 2).
So, the given expression is equal to -4/(x^2 + 2) after expressing it in partial fractions.