In Exercises 43–46, perform each long division and write the partial fraction decomposition of the remainder term. (x^5+2)/(x^2-1)

Key Concepts

Long Division of Polynomials

Long division of polynomials 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.

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 rational functions. The process involves factoring the denominator and expressing the function as a sum of fractions whose denominators are the factors of the original denominator, allowing for easier manipulation and integration.

Remainder Theorem

The Remainder Theorem states that when a polynomial f(x) is divided by a linear divisor of the form (x - c), the remainder of this division is f(c). In the context of polynomial long division, the remainder can be expressed as a polynomial of lower degree than the divisor, which is essential for performing partial fraction decomposition and understanding the overall division process.