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Chapter 12: Problem 5

Write the partial fraction decomposition for the expression. $$ \frac{4 x-13}{x^{2}-3 x-10} $$

Short Answer

Expert verified
The partial fraction decomposition for the given expression is \[\frac{4x-13}{(x-5)(x+2)} = \frac{3}{x-5} + \frac{1}{x+2}\]

Step by step solution

01

Factorize Denominator

Factorize the denominator of the given function. The denominator is a quadratic equation \(x^{2}-3x-10\). To factorize it, we find two numbers that add up to -3 (the coefficient of \(x\)) and multiply to -10 (the constant term). The numbers -5 and 2 fit these criteria and hence, the factorization of the denominator is \((x-5)(x+2)\). So the expression becomes: \[\frac{4x-13}{(x-5)(x+2)}\]
02

Set Up Partial Fraction Decomposition

Set up the partial fractions decomposition. This is achieved by expressing as a sum of two fractions where the denominators are the factors of the original denominator and each has a constant \(A\) and \(B\), unknown at this point, as a numerator. The partial fraction decomposition is set up as: \[\frac{4x-13}{(x-5)(x+2)} = \frac{A}{x-5} + \frac{B}{x+2}\]
03

Solve for A and B

To solve for the constants \(A\) and \(B\), get rid of the denominators by multiplying through by \((x-5)(x+2)\). This gives: \[4x-13 = A(x+2) + B(x-5)\]On expanding the expressions on the right hand side and grouping like terms together, this equation becomes: \[4x - 13 = (A + B) x + 2A -5B\]Setting the coefficients equal gives the two equations: \(A + B = 4\) and \(2A - 5B = -13\). Solving this system of equations yields \(A = 3\) and \(B = 1\). This gives us the final partial fraction decomposition: \[\frac{4x-13}{(x-5)(x+2)} = \frac{3}{x-5} + \frac{1}{x+2}\]

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Most popular questions from this chapter

Find the indefinite integral (a) using the integration table and (b) using the specified method. Integral \mathrm{Method } $$ \int x^{4} \ln x d x \quad \text { Integration by parts } $$

Use the error formulas to find \(n\) such that the error in the approximation of the definite integral is less than \(0.0001\) using (a) the Trapezoidal Rule and (b) Simpson's Rule. $$ \int_{0}^{1} x^{3} d x $$

Explain why the integral is improper and determine whether it diverges or converges. Evaluate the integral if it converses. $$ \int_{0}^{2} \frac{1}{(x-1)^{2 / 3}} d x $$

Use a program similar to the Simpson's Rule program on page 906 to approximate the integral. Use \(n=100\). $$ \int_{2}^{5} 10 x^{2} e^{-x} d x $$

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.) $$ \int_{0}^{2} \frac{1}{\sqrt{1+x^{3}}} d x, n=4 $$
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