Find the sum of the terms.
$\frac{x}{x^2+3 x+2}+\frac{3}{x+1}$
The numerator of the simplified sum is [ ] .
To simplify the expression \(\frac{x}{x^2 + 3x + 2} + \frac{3}{x + 1}\), we first analyze the structure of the denominator in the first fraction. The quadratic polynomial \(x^2 + 3x + 2\) can be factored by finding two numbers that multiply to 2 and add to 3. These numbers are 1 and 2. Therefore, \(x^2 + 3x + 2 = (x + 1)(x + 2)\).
With the denominator factored, the expression becomes \(\frac{x}{(x + 1)(x + 2)} + \frac{3}{x + 1}\). To add these two fractions, we need a common denominator, which is \((x + 1)(x + 2)\). The second fraction needs to be adjusted by multiplying its numerator and denominator by \((x + 2)\):
\[ \frac{3}{x + 1} = \frac{3(x + 2)}{(x + 1)(x + 2)} = \frac{3x + 6}{(x + 1)(x + 2)}. \]
Now, we add the two fractions together:
\[ \frac{x}{(x + 1)(x + 2)} + \frac{3x + 6}{(x + 1)(x + 2)} = \frac{x + 3x + 6}{(x + 1)(x + 2)}. \]
Combining the like terms in the numerator gives:
\[ \frac{4x + 6}{(x + 1)(x + 2)}. \]
The resulting numerator is \(4x + 6\).
In engineering and physical sciences, simplifying complex algebraic expressions is a fundamental skill. For instance, in electrical circuit analysis, finding the total equivalent resistance of parallel resistors often requires adding rational expressions. If you have resistances that depend on a variable component \(x\), you must simplify the sum of their conductance expressions to design a circuit that behaves predictably across various load conditions. Learning how to add these fractions accurately ensures your final design parameters are correct.