Final answer:
To find the partial fraction decomposition, factor the denominator and equate numerators to solve for the coefficients.
Step-by-step explanation:
To find the partial fraction decomposition of the rational expression (3x²-2x+4)/(x²(2-x)), we need to factor the denominator completely. Factoring x²(2-x), we get x²(x-2). Now, we can write the expression as:
(3x²-2x+4)/(x²(x-2)) = A/x + B/x² + C/(x-2)
To find the values of A, B, and C, we can multiply each side of the equation by the common denominator, x²(x-2). After simplifying, we equate the numerators:
3x²-2x+4 = A(x)(x-2) + B(x-2) + C(x²)
Now we can solve for A, B, and C by comparing the coefficients of like terms. Once we have the values of A, B, and C, we can write the partial fraction decomposition of the rational expression.