Perform each long division and write the partial fraction decomposition of the remainder term. (x⁵+2)/(x²-1)

Verified step by step guidance

1. Identify the dividend and divisor for the long division: the dividend is \(x^{5} + 2\) and the divisor is \(x^{2} - 1\).
2. Set up the long division by dividing the leading term of the dividend, \(x^{5}\), by the leading term of the divisor, \(x^{2}\), to find the first term of the quotient.
3. Multiply the entire divisor \(x^{2} - 1\) by the term found in the previous step and subtract this product from the current dividend to find the new remainder.
4. Repeat the division process with the new remainder: divide its leading term by \(x^{2}\), multiply the divisor by this term, and subtract again, continuing until the degree of the remainder is less than the degree of the divisor.
5. Express the original rational expression as the quotient plus the remainder over the divisor, then write the partial fraction decomposition of the remainder term by factoring the divisor \(x^{2} - 1\) into \((x - 1)(x + 1)\) and setting up the decomposition accordingly.

Key Concepts

Polynomial Long Division

Polynomial long division is a method used to divide a polynomial by another polynomial of lower degree, similar to numerical long division. It helps to express the division as a quotient plus a remainder over the divisor, which is essential for simplifying rational expressions.

Partial Fraction Decomposition

Partial fraction decomposition breaks down a rational expression into a sum of simpler fractions with denominators that are factors of the original denominator. This technique is useful for integrating rational functions or simplifying expressions after division.

Factoring Quadratic Expressions

Factoring quadratic expressions involves rewriting a quadratic polynomial as a product of simpler binomials. Recognizing that x² - 1 factors as (x - 1)(x + 1) is crucial for setting up the partial fractions correctly in the decomposition process.