In Exercises 9–42, write the partial fraction decomposition of each rational expression. x²+2x+7/x(x − 1)²

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1. Identify the denominator factors and their powers. Here, the denominator is $x(x-1{)}^2$, which factors into $x$ and $\mathrm{(x\; -\; 1)^2}$.
2. Set up the form of the partial fraction decomposition. Since $x$ is a linear factor and $\mathrm{(x\; -\; 1)^2}$ is a repeated linear factor, write the decomposition as $\backslash frac\{A\}\{x\}\; +\; \backslash frac\{B\}\{x\; -\; 1\}\; +\; \backslash frac\{C\}\{(x\; -\; 1)^2\}$, where A, B, and C are constants to be determined.
3. Write the equation equating the original rational expression to the sum of the partial fractions: $\frac{x^2\; +\; 2x\; +\; 7}{x(x\; -\; 1)^2}=\; \backslash frac\{A\}\{x\}\; +\; \backslash frac\{B\}\{x\; -\; 1\}\; +\; \backslash frac\{C\}\{(x\; -\; 1)^2\}$.
4. Multiply both sides of the equation by the common denominator $x(x\; -\; 1)^2$ to clear the denominators, resulting in: $x^2\; +\; 2x\; +\; 7\; =\; A(x\; -\; 1)^2\; +\; Bx(x\; -\; 1)\; +\; Cx$.
5. Expand the right-hand side and collect like terms in powers of $x$. Then, equate the coefficients of corresponding powers of $x$ on both sides to form a system of equations to solve for A, B, and C.

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 denominators that are factors of the original denominator. This technique is useful for integration and solving equations.

Factoring and Repeated Factors

Factoring the denominator into linear and repeated factors is crucial for setting up the correct form of partial fractions. Repeated factors require terms with increasing powers in the decomposition, such as (x - 1) and (x - 1)^2.