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Chapter 0: Problem 57

Add or subtract as indicated. $$\frac{4 x^{2}+x-6}{x^{2}+3 x+2}-\frac{3 x}{x+1}+\frac{5}{x+2}$$

Short Answer

Expert verified
The solution is \(\frac{x^2 - 4x - 1}{(x+1)(x+2)}\)

Step by step solution

01

Simplify the denominator

The first step will involve simplifying the denominator of the fractions, if possible. Observing the first fraction \(\frac{4x^2+x-6}{x^2+3x+2}\), its denominator \(x^2+3x+2\) can be factored out into \((x+1)(x+2)\). The other denominators are \(x+1\) and \(x+2\), which cannot be simplified further.
02

Obtain the Common Denominator

Next, find a common denominator among the three fractions. The least common multiple (LCM) of \((x+1)(x+2), x+1,\) and \(x+2\) is \((x+1)(x+2)\). Thus we need to re-write the fractions as follows to have a common denominator: \[ \frac{4x^2 + x - 6}{(x+1)(x+2)} - \frac{3x(x+2)}{(x+1)(x+2)} + \frac{5(x+1)}{(x+1)(x+2)}\]
03

Perform the Additions and Subtractions

Now perform the additions and subtractions in the numerator and simplify where possible. This gives: \[(4x^2 + x - 6) - (3x^2 + 6x) + (5x + 5)\] Simplifying this gives \(x^2 - 4x -1\] And so the final answer can be written as \[\frac{x^2 - 4x - 1}{(x+1)(x+2)}\]

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