Question

$$ \frac{5 x^{2}+9 x-1}{(2 x+3)\left(x^{2}+5 x+2\right)} $$

   $$
\frac{5 x^{2}+9 x-1}{(2 x+3)\left(x^{2}+5 x+2\right)}
$$

Engineering Mathematics

Engineering Mathematics

Stroud, K. A. Booth,… 7th Edition

Chapter 8, Problem 21

Instant Answer

Solved by Expert M Hassan Anwar

02/13/2022

Step 1

Step 1: We are given the expression $$ \frac{5 x^{2}+9 x-1}{(2 x+3)\left(x^{2}+5 x+2\right)} $$ We want to express this as a sum of partial fractions of the form $\frac{A}{2x+3} + \frac{Bx+C}{x^2+5x+2}$. Show more…

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$$ \frac{5 x^{2}+9 x-1}{(2 x+3)\left(x^{2}+5 x+2\right)} $$

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Key Concepts

Rational Functions

A rational function is a ratio of two polynomials. Understanding these functions involves analyzing the degrees of the numerator and denominator, which helps determine properties like end behavior, potential asymptotes, and whether the function is proper (where the degree of the numerator is less than the degree of the denominator) or improper.

Partial Fraction Decomposition

Partial fraction decomposition is a method used to rewrite a rational function as a sum of simpler fractions. This technique is particularly helpful for integration and solving differential equations. It involves expressing the original function in terms of fractions with simpler denominators, often related to the factors of the original denominator.

Polynomial Factorization

Polynomial factorization involves expressing a polynomial as a product of its factors, which could be linear or irreducible quadratic factors. In the context of rational functions, factorizing the denominator simplifies the process of partial fraction decomposition by revealing the basic building blocks onto which the function can be decomposed.

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