Problem 57 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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.

Step 2: 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: \[ \frac{4x^2 + x - 6}{(x+1)(x+2)} - \frac{3x(x+2)}{(x+1)(x+2)} + \frac{5(x+1)}{(x+1)(x+2)}\]

Step 3: 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 \[\frac{x^2 - 4x - 1}{(x+1)(x+2)}\]