Question

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

Accepted Answer

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: $$\frac{2x^5$ + 3x^4$ - 3x^3$ - 2x^2$ + x}{2x^2$ + 5x + 2} = 	ext{quotient} + \frac{	ext{remainder}}{2x^2$ + 5x + 2}. $$Next, factor the denominator $$2x^2 + 5x + 2$ into linear factors. To factor, find two numbers that multiply to 2$$ 	imes 2 = 4$$ and add to 5. Then rewrite and factor by grouping. Set up the partial fraction decomposition for the proper rational expression $$\frac{	ext{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.

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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Key Concepts

Partial Fraction Decomposition

Polynomial Long Division

Factoring Quadratic Expressions