Write the partial fraction decomposition of each rational expression. x+4/x² (x²+4)

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Identify the given rational expression: \(\frac{x+4}{x^{2}(x^{2}+4)}\).

Recognize the factors in the denominator: \(x^{2}\) is a repeated linear factor (since \(x\) is linear, repeated twice), and \(x^{2}+4\) is an irreducible quadratic factor.

Set up the partial fraction decomposition form. For the repeated linear factor \(x^{2}\), include terms with denominators \(x\) and \(x^{2}\), and for the irreducible quadratic \(x^{2}+4\), include a term with a linear numerator:

\[\frac{x+4}{x^{2}(x^{2}+4)} = \frac{A}{x} + \frac{B}{x^{2}} + \frac{Cx + D}{x^{2} + 4}.\]

Multiply both sides of the equation by the common denominator \(x^{2}(x^{2}+4)\) to clear the denominators:

\[x + 4 = A \cdot x (x^{2} + 4) + B (x^{2} + 4) + (Cx + D) x^{2}.\]

Expand the right side and then collect like terms by powers of \(x\). This will allow you to equate coefficients of corresponding powers of \(x\) on both sides to form a system of equations to solve for \(A\), \(B\), \(C\), and \(D\).

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주요 개념

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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 them as products of simpler polynomials. Recognizing factors like quadratic expressions and their irreducibility over the reals is crucial for setting up the correct form of partial fractions.