Find the partial fraction decomposition for each rational expression. See Examples 1–4. (2x⁵ + 3x⁴ - 3x³ - 2x² + x)/(2x² + 5x + 2)

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First, observe that the degree of the numerator (which is 5) is greater than the degree of the denominator (which is 2). This means we need to perform polynomial long division before decomposing into partial fractions.

Perform polynomial long division by dividing the numerator \(2x^5 + 3x^4 - 3x^3 - 2x^2 + x\) by the denominator \(2x^2 + 5x + 2\). This will give a quotient polynomial and a remainder polynomial.

Express the original rational expression as the sum of the quotient polynomial plus the remainder over the original denominator: \(\frac{2x^5 + 3x^4 - 3x^3 - 2x^2 + x}{2x^2 + 5x + 2} = \text{quotient} + \frac{\text{remainder}}{2x^2 + 5x + 2}\).

Next, factor the denominator \(2x^2 + 5x + 2\) into linear factors. To factor, find two numbers that multiply to \(2 \times 2 = 4\) and add to 5. Then rewrite and factor by grouping.

Set up the partial fraction decomposition for the proper rational expression \(\frac{\text{remainder}}{(2x + 1)(x + 2)}\) as \(\frac{A}{2x + 1} + \frac{B}{x + 2}\), where \(A\) and \(B\) are constants to be determined by multiplying both sides by the denominator and equating coefficients.

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주요 개념

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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.

Polynomial Long Division

When the degree of the numerator is equal to or greater than the degree of the denominator, polynomial long division is performed first. This process divides the polynomials to rewrite the expression as a polynomial plus a proper fraction, which is necessary before applying partial fraction decomposition.

Factoring Quadratic Expressions

Factoring the denominator into linear or irreducible quadratic factors is essential for setting up the partial fractions correctly. Recognizing how to factor quadratics like 2x^2 + 5x + 2 into (2x + 1)(x + 2) allows the decomposition into simpler terms with unknown coefficients to solve.

Find the partial fraction decomposition for each rational expression. See Examples 1–4. (2x5 + 3x4 - 3x3 - 2x2 + x)/(2x2 + 5x + 2)