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)

Verified step by step guidance

1. Step 1: Set up the long division by writing \(x^4 - x^2 + 2\) as the dividend and \(x^3 - x^2\) as the divisor.
2. Step 2: Divide the first term of the dividend \(x^4\) by the first term of the divisor \(x^3\) to get the first term of the quotient, which is \(x\).
3. Step 3: Multiply the entire divisor \(x^3 - x^2\) by the first term of the quotient \(x\) and subtract the result from the dividend.
4. Step 4: Bring down the next term from the original dividend if necessary and repeat the division process with the new polynomial obtained after subtraction.
5. Step 5: Once the degree of the remainder is less than the degree of the divisor, express the remainder as a partial fraction decomposition if possible.

Key Concepts

Polynomial Long Division

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.

Remainder and Quotient

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

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.