Find the partial fraction decomposition for each rational expression. See Examples 1–4. (x²)/(x² + 2x + 1)

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First, recognize that the denominator \(x^2 + 2x + 1\) can be factored. Factor it as \(\left(x + 1\right)^2\) because it is a perfect square trinomial.

Rewrite the rational expression using the factored denominator: \(\frac{x^2}{\left(x + 1\right)^2}\).

Set up the partial fraction decomposition form for a repeated linear factor \(\left(x + 1\right)^2\). The general form is: \(\frac{A}{x + 1} + \frac{B}{\left(x + 1\right)^2}\), where \(A\) and \(B\) are constants to be determined.

Multiply both sides of the equation by the common denominator \(\left(x + 1\right)^2\) to clear the fractions: \(x^2 = A(x + 1) + B\).

Expand the right side and collect like terms: \(x^2 = A x + A + B\). Then, equate the coefficients of corresponding powers of \(x\) on both sides to form a system of equations to solve for \(A\) and \(B\).

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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. This technique is especially useful for integrating rational functions or simplifying complex expressions. It involves breaking down a fraction into components with denominators that are factors of the original denominator.

Factoring Quadratic Expressions

Factoring quadratic expressions involves rewriting a quadratic polynomial as a product of two binomials. For example, x^2 + 2x + 1 factors to (x + 1)^2. Recognizing and factoring the denominator is essential in partial fraction decomposition to determine the form of the simpler fractions.

Handling Repeated Factors in Denominators

When the denominator has repeated factors, such as (x + 1)^2, the partial fraction decomposition must include terms for each power of the repeated factor. For (x + 1)^2, the decomposition includes terms with denominators (x + 1) and (x + 1)^2, each with its own numerator to solve for.

Find the partial fraction decomposition for each rational expression. See Examples 1–4. (x2)/(x2 + 2x + 1)