In Exercises 43–46, perform each long division and write the partial fraction decomposition of the remainder term. (x^4-x^2+2)/(x^3-x^2)

Polynomial long division is a method used to divide a polynomial by another polynomial of lower degree. It involves dividing the leading term of the dividend by the leading term of the divisor, multiplying the entire divisor by this result, and subtracting it from the dividend. This process is repeated until the degree of the remainder is less than the degree of the divisor.

In polynomial division, the result consists of a quotient and a remainder. The quotient is the polynomial obtained from the division process, while the remainder is what is left over after the division. The remainder can be expressed as a fraction over the divisor, and it is crucial for further analysis, such as partial fraction decomposition.

Partial fraction decomposition is a technique used to express a rational function as a sum of simpler fractions. This is particularly useful when integrating or simplifying expressions. The process involves breaking down the remainder term into fractions whose denominators are the factors of the original polynomial, allowing for easier manipulation and integration.