Divide.
$\frac{x+1}{x^2+5 x+6} wedge \frac{3 x+3}{x^2+2 x-3}$
Simplify your answer as much as possible.
We are asked to simplify the division of two rational expressions:
\[\frac{x+1}{x^2+5x+6} \div \frac{3x+3}{x^2+2x-3}\]
A rational expression is a fraction where the numerator and denominator are polynomials. Dividing by a fraction is equivalent to multiplying by its reciprocal. Our goal is to factor each component and simplify the resulting expression.
To divide by a fraction, we multiply by its reciprocal:
\[\frac{x+1}{x^2+5x+6} \cdot \frac{x^2+2x-3}{3x+3}\]
Next, we factor each polynomial involved:
Substituting these factored forms into our expression gives:
\[\frac{x+1}{(x+2)(x+3)} \cdot \frac{(x+3)(x-1)}{3(x+1)}\]
Now, we look for common factors in the numerator and denominator:
\[\frac{(x+1)(x+3)(x-1)}{(x+2)(x+3) \cdot 3(x+1)}\]
We can cancel the common factors \((x+1)\) and \((x+3)\), provided they are not equal to zero. This leaves us with:
\[\frac{x-1}{3(x+2)}\]
This is the most simplified form of the given expression.
Simplifying rational expressions is fundamental in fields like physics and engineering, where complex formulas are reduced to manageable forms. For instance, when analyzing the resistance in a parallel electrical circuit or calculating the velocity of an object over time, you often encounter rational expressions. Reducing these expressions allows for easier calculations, better insight into the relationship between variables, and faster analysis of real-world phenomena.