In Exercises 43–46, perform each long division and write the partial fraction decomposition of the remainder term. (x⁴-x²+2)/(x³-x²)

Verified step by step guidance

1. Identify the dividend and divisor for the long division: dividend is $x{x}^4-x^2+2$ and divisor is $x{x}^3-x^2$.
2. Set up the long division by dividing the leading term of the dividend, $x^4$, by the leading term of the divisor, $x^3$, to find the first term of the quotient.
3. Multiply the entire divisor by this first term of the quotient and subtract the result from the dividend to find the new remainder.
4. Repeat the division process with the new remainder: divide its leading term by the leading term of the divisor, multiply the divisor by this term, and subtract again 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 and expressing the remainder as a sum of simpler fractions with unknown coefficients.

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 terms, multiplying, subtracting, and bringing down the next term repeatedly until the remainder has a degree less than the divisor. This process helps simplify complex rational expressions.

Partial Fraction Decomposition

Partial fraction decomposition breaks a rational expression into a sum of simpler fractions with denominators that are factors of the original denominator. This technique is useful for integration and solving equations. The remainder from polynomial division is expressed as a fraction with the original divisor.

Factoring Polynomials

Factoring polynomials involves expressing a polynomial as a product of its factors, which can be linear or quadratic. Factoring the denominator is essential for partial fraction decomposition, as it determines the form of the simpler fractions. Recognizing common factors and special products aids in this process.