Write the partial fraction decomposition of each rational expression. x²+2x+7/x(x − 1)²

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Identify the form of the denominator and write the general form of the partial fraction decomposition. Since the denominator is \(x(x - 1)^2\), the decomposition will include terms for the linear factor \(x\) and the repeated linear factor \((x - 1)^2\).

Set up the partial fraction decomposition as: \(\frac{A}{x} + \frac{B}{x - 1} + \frac{C}{(x - 1)^2}\), where \(A\), \(B\), and \(C\) are constants to be determined.

Multiply both sides of the equation by the common denominator \(x(x - 1)^2\) to clear the fractions, resulting in: \(x^2 + 2x + 7 = A(x - 1)^2 + Bx(x - 1) + Cx\).

Expand the right-hand side by applying the distributive property and simplifying each term: expand \((x - 1)^2\), multiply terms, and combine like terms.

Equate the coefficients of corresponding powers of \(x\) on both sides of the equation to form a system of linear equations in \(A\), \(B\), and \(C\). Solve this system to find the values of \(A\), \(B\), and \(C\).

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Partial Fraction Decomposition

Partial fraction decomposition is a method used to express a rational function as a sum of simpler fractions with denominators that are factors of the original denominator. This technique simplifies integration and other algebraic operations by breaking down complex fractions into manageable parts.

Factorization of Denominators

Understanding how to factor the denominator is essential for setting up partial fractions. In this problem, the denominator x(x − 1)² includes a linear factor and a repeated linear factor, which affects the form of the decomposition by requiring terms for each power of the repeated factor.

Setting Up and Solving Equations for Coefficients

After expressing the rational expression as a sum of partial fractions, you equate the numerators and solve for unknown coefficients. This involves multiplying both sides by the common denominator and comparing coefficients of corresponding powers of x to form a system of equations.

Write the partial fraction decomposition of each rational expression. x2+2x+7/x(x − 1)2