In Exercises 9–42, write the partial fraction decomposition of each rational expression. 6x^2-x+1/(x^3 + x²+x+1)
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<Step 1: Identify the type of partial fraction decomposition needed.>
<Step 2: Factor the denominator \(x^3 + x^2 + x + 1\).>
<Step 3: Set up the partial fraction decomposition based on the factors of the denominator.>
<Step 4: Write the expression as a sum of fractions with unknown coefficients.>
<Step 5: Solve for the unknown coefficients by equating coefficients or substituting values.>
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Rational Expressions
A rational expression is a fraction where both the numerator and the denominator are polynomials. Understanding rational expressions is crucial for performing operations such as addition, subtraction, multiplication, and division, as well as for decomposing them into simpler fractions. In this context, recognizing the structure of the given expression helps in identifying how to break it down into partial fractions.
Partial fraction decomposition is a technique used to express a rational function as a sum of simpler fractions. This method is particularly useful for integrating rational functions or simplifying complex expressions. The process involves factoring the denominator and expressing the original fraction as a sum of fractions whose denominators are the factors of the original denominator.
Factoring polynomials is the process of breaking down a polynomial into simpler components, or factors, that when multiplied together yield the original polynomial. This is essential in partial fraction decomposition, as the first step is to factor the denominator completely. Understanding how to factor polynomials, including recognizing irreducible quadratics and linear factors, is vital for successfully applying the decomposition technique.