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 structure of the expression. We have a sum of two rational expressions where the first denominator is a quadratic polynomial. To combine these fractions, we need to find a common denominator.
The quadratic expression in the denominator of the first term is \(x^2+3x+2\). This can be factored into a product of two linear factors by finding two numbers that multiply to 2 and add to 3. Those numbers are 1 and 2. Therefore, we can write:
\[x^2+3x+2 = (x+1)(x+2).\]
Now the expression becomes:
\[\frac{x}{(x+1)(x+2)} + \frac{3}{x+1}.\]
To add these two fractions, we need them to have the same denominator, which is \((x+1)(x+2)\). We multiply the numerator and denominator of the second term by \((x+2)\):
\[\frac{3}{x+1} = \frac{3(x+2)}{(x+1)(x+2)} = \frac{3x+6}{(x+1)(x+2)}.\]
Now we combine the two fractions:
\[\frac{x}{(x+1)(x+2)} + \frac{3x+6}{(x+1)(x+2)} = \frac{x + 3x + 6}{(x+1)(x+2)}.\]
Simplifying the numerator, we combine the like terms:
\[x + 3x + 6 = 4x + 6.\]
Thus, the simplified sum of the expressions is \(\frac{4x+6}{(x+1)(x+2)}\). The numerator of this simplified fraction is \(4x+6\). This matches the third option in the provided list.
Understanding how to simplify rational expressions is essential for solving more complex algebraic equations. For example, if you are designing a system where two different components contribute to a total, like two pipes filling a tank at different rates or two resistors in a circuit, you often need to combine these rates into a single expression to find the total rate of work. Mastering these algebraic manipulations ensures you can combine and analyze system dynamics accurately in physics and engineering applications.