In Exercises 9–42, write the partial fraction decomposition of each rational expression. 1/x(x-1)

Verified step by step guidance

1. Identify the form of the rational expression. Here, the denominator is a product of two linear factors: $x(x-1)$.
2. Set up the partial fraction decomposition by expressing the rational expression as a sum of fractions with unknown constants in the numerators over each linear factor: $\backslash frac\{1\}\{x(x-1)\}\; =\; \backslash frac\{A\}\{x\}\; +\; \backslash frac\{B\}\{x-1\}$, where $A$ and $B$ are constants to be determined.
3. Multiply both sides of the equation by the common denominator $x(x-1)$ to clear the denominators: $1\; =\; A(x-1)\; +\; Bx$.
4. Expand the right side: $1\; =\; A\; x\; -\; A\; +\; B\; x$, then combine like terms: $1\; =\; (A\; +\; B)\; x\; -\; A$.
5. Equate the coefficients of corresponding powers of $x$ on both sides to form a system of equations: For the $x$ term, $0\; =\; A\; +\; B$; for the constant term, $1\; =\; -A$. These equations can be solved to find $A$ and $B$.

Key Concepts

Rational Expressions

A rational expression is a fraction where both the numerator and denominator are polynomials. Understanding how to manipulate these expressions is essential for simplifying, factoring, and decomposing them into partial fractions.

Partial Fraction Decomposition

Partial fraction decomposition involves expressing a complex rational expression as a sum of simpler fractions with simpler denominators. This technique is useful for integration and solving equations involving rational expressions.

Factoring Polynomials

Factoring polynomials means rewriting a polynomial as a product of its factors. Recognizing and factoring denominators into linear or irreducible quadratic factors is crucial for setting up the correct form of partial fractions.