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Chapter 12: Problem 6

Write the partial fraction decomposition for the expression. $$ \frac{7 x+5}{6\left(2 x^{2}+3 x+1\right)} $$

Short Answer

Expert verified
The partial fraction decomposition for the given expression is \(\frac{7x + 5}{6(2x^2 + 3x + 1)}\). Because the quadratic term in the denominator cannot be further factorized, only one constant is obtained.

Step by step solution

01

Extract the factors of the quadratic term in the denominator

This step requires you to factorize the quadratic term. However, the quadratic term \(2x^2 + 3x + 1\) cannot be further factorized because it has only one unique root. So, we proceed without factorizing the denominator.
02

Express the fraction as partial fractions

A fraction can be expressed as the sum of its partial fractions. Since the denominator is a quadratic having only one root, we express the fraction in the following form: \(\frac{7x + 5}{6(2x^2 + 3x + 1)} = \frac{A}{2x^2 + 3x + 1}\)
03

Solve for the constant in the partial fraction

To solve for the constant \(A\), we multiply through by \(2x^2 + 3x + 1\), yielding \(7x + 5 = A\). As there is no x term in the equation, this is a valid standing equation that always holds true. Thus, we can solve for \(A\) by substituting the root of denominator (values of x that make denominator zero) and then solve for \(A\)

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Most popular questions from this chapter

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges. $$ \int_{1 / 2}^{\infty} \frac{1}{\sqrt{2 x-1}} d x $$

Approximate the integral using (a) the Trapezoidal Rule and (b) Simpson's Rule for the indicated value of \(n\). (Round your answers to three significant digits.) $$ \int_{0}^{2} \sqrt{1+x^{3}} d x, n=4 $$

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