Divide.
$\frac{x^2+4 x+3}{x+2} \div \frac{3 x+3}{x-3}$
We are tasked with dividing the rational expression \(\frac{x^2+4x+3}{x+2}\) by \(\frac{3x+3}{x-3}\). Dividing by a rational expression is equivalent to multiplying by its reciprocal. Therefore, we rewrite the expression as:
\[\frac{x^2+4x+3}{x+2} \cdot \frac{x-3}{3x+3}\]
To simplify this product, we identify the domain restrictions. The original denominator \(x+2\) cannot be zero, so \(x
eq -2\). The denominator of the second fraction \(3x+3\) cannot be zero, so \(x
eq -1\). Additionally, the numerator of the second fraction \(3x+3\) being in the original denominator of the division indicates that the original expression \(\frac{3x+3}{x-3}\) cannot be zero, meaning \(3x+3
eq 0\), reinforcing \(x
eq -1\). For the simplified result to be valid, we ensure these constraints are maintained.
Next, we factor the quadratic and linear expressions to facilitate the identification of common terms.
The quadratic expression in the first numerator, \(x^2+4x+3\), can be factored as:
\[x^2+4x+3 = (x+1)(x+3)\]
The expression in the second denominator, \(3x+3\), can be factored by taking out the common factor of 3:
\[3x+3 = 3(x+1)\]
Substituting these factored forms back into our multiplication problem, we have:
\[\frac{(x+1)(x+3)}{x+2} \cdot \frac{x-3}{3(x+1)}\]
Now, we simplify the expression by canceling the common factor \((x+1)\) that appears in both the numerator and the denominator.
\[\frac{\cancel{(x+1)}(x+3)}{x+2} \cdot \frac{x-3}{3\cancel{(x+1)}}\]
After canceling the factor of \((x+1)\), the expression becomes:
\[\frac{(x+3)(x-3)}{3(x+2)}\]
This can be further simplified by multiplying the factors in the numerator. Using the difference of squares identity \((a+b)(a-b) = a^2 - b^2\), we have \((x+3)(x-3) = x^2 - 9\).
Thus, the final simplified expression is \(\frac{x^2-9}{3(x+2)}\), given the restrictions \(x
eq -2, -1, 3\).
The final simplified form of the given expression is:
\[\boxed{\frac{x^2-9}{3x+6}}\]
where the expression is defined for \(x
eq -2, -1, 3\).
In structural engineering, when analyzing the distribution of force across different load-bearing beams represented by rational functions, you often need to simplify complex equations of forces. By representing the ratio of load distributions as rational expressions, you can simplify the system by factoring out common terms (like beam material properties or shared geometry constraints), making it much easier to solve for the unknown stress values in complex architectural designs.