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Chapter 8: Problem 54

Write the form of the partial fraction decomposition of the rational expression. Do not solve for the constants. $$\frac{x-2}{x^{2}+4 x+3}$$

Short Answer

Expert verified
The form of the partial fraction decomposition of the given rational expression is \(\frac{x-2}{x^{2}+4 x+3}\) = \(\frac{A}{x+1}\) + \(\frac{B}{x+3}\)

Step by step solution

01

Factoring the denominator

Factor the quadratic expression in the denominator. \(x^{2}+4 x+3\) can be factored into \((x+1)(x+3)\)
02

Set up the partial fraction decomposition

Set up the partial fraction decomposition. The general form of the decomposition will be: \(\frac{x-2}{x^{2}+4 x+3}\) = \(\frac{A}{x+1}\) + \(\frac{B}{x+3}\) where A and B are constants to be determined.

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Most popular questions from this chapter

Use an inverse matrix to solve (if possible) the system of linear equations. $$\left\\{\begin{array}{l} 4 x-y+z=-5 \\ 2 x+2 y+3 z=10 \\ 5 x-2 y+6 z=1 \end{array}\right.$$

Determine whether the statement is true or false. Justify your answer. If a system of linear equations has two distinct solutions, then it has an infinite number of solutions.

Consider the system of equations. $$\left\\{\begin{array}{l} y=b^{x} \\ y=x^{b} \end{array}\right.$$ (a) Use a graphing utility to graph the system of equations for \(b=2\) and \(b=4\) (b) For a fixed value of \(b > 1,\) make a conjecture about the number of points of intersection of the graphs in part (a).

(A) find the determinant of \(A,\) (b) find \(A^{-1},\) (c) find \(\operatorname{det}\left(A^{-1}\right),\) and (d) compare your results from parts (a) and (c). Make a conjecture based on your results. $$A=\left[\begin{array}{ll} 5 & -1 \\ 2 & -1 \end{array}\right]$$

Determine whether the statement is true or false. Justify your answer. When the product of two square matrices is the identity matrix, the matrices are inverses of one another.
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