What is the numerator of the simplified sum?
$\frac{x}{x^2+3x+2}+\frac{3}{x+1}$
To simplify the expression \(\frac{x}{x^2+3x+2} + \frac{3}{x+1}\), we first analyze the denominator of the first fraction. The quadratic expression \(x^2+3x+2\) can be factored by finding two numbers that multiply to 2 and add up to 3. These numbers are 1 and 2. Therefore, \(x^2+3x+2 = (x+1)(x+2)\).
Now rewrite the original expression using the factored form:
\[\frac{x}{(x+1)(x+2)} + \frac{3}{x+1}\]
To add these fractions, we need a common denominator. The least common denominator is \((x+1)(x+2)\). The first fraction already has this denominator, but the second fraction needs to be adjusted. We multiply the numerator and the denominator of the second fraction by \((x+2)\):
\[\frac{3}{x+1} = \frac{3(x+2)}{(x+1)(x+2)}\]
Now combine the two fractions over the common denominator:
\[\frac{x}{(x+1)(x+2)} + \frac{3(x+2)}{(x+1)(x+2)} = \frac{x + 3(x+2)}{(x+1)(x+2)}\]
Next, distribute the 3 in the numerator:
\[x + 3(x+2) = x + 3x + 6 = 4x + 6\]
So, the simplified expression is \(\frac{4x+6}{(x+1)(x+2)}\). The numerator of this simplified sum is \(4x+6\).
In real-world applications, such as electrical circuit analysis or managing resource distributions, we often encounter complex rational expressions that need to be simplified to understand the system's behavior. By combining fractions with a common denominator, we can reduce complex relationships into a single manageable term. For example, if two separate components in a circuit contribute to the total impedance, represented by fractions of variables, simplifying them allows us to find the combined effect much more easily.