Divide.
$\frac{4 x-4}{x-3} \div \frac{x^2+x-2}{x^2-9}$
Simplify your answer as much as possible.
To simplify the given rational expression \(\frac{4x - 4}{x - 3} \div \frac{x^2 + x - 2}{x^2 - 9}\), we must first recall that dividing by a fraction is equivalent to multiplying by its reciprocal. Thus, we rewrite the expression as:
\[ \frac{4x - 4}{x - 3} \cdot \frac{x^2 - 9}{x^2 + x - 2} \]
Before performing the multiplication, we must ensure the expression is well-defined. The original expression is defined as long as the denominators are non-zero. Specifically, \(x - 3
eq 0\), \(x^2 - 9
eq 0\), and \(x^2 + x - 2
eq 0\). This leads to constraints \(x
eq 3\), \(x
eq -3\), \(x
eq 1\), and \(x
eq -2\).
Now, we factor each polynomial within the numerators and denominators to identify common terms:
Substituting these factored forms back into our expression gives:
\[ \frac{4(x - 1)}{x - 3} \cdot \frac{(x - 3)(x + 3)}{(x + 2)(x - 1)} \]
With the expression fully factored, we can now cancel common factors in the numerator and the denominator. We see that \((x - 1)\) appears in both the numerator and denominator, and \((x - 3)\) also appears in both. Canceling these terms, we are left with:
\[ \frac{4(x + 3)}{x + 2} \]
This simplified fraction represents the original expression, under the assumption that the variables do not make the initial denominators zero.
Simplifying rational expressions is an essential skill in algebra that appears frequently in fields like engineering and physics. For instance, when designing a circuit and calculating total impedance, you often end up with complex rational expressions involving various resistances and capacitances. Simplifying these early on can make solving for a specific variable, like current or frequency, much more manageable, reducing the chance of calculation errors as you analyze your system's behavior.