Simplify.
$\frac{\frac{x^2}{x-1}}{\frac{3 x^2}{x+4}}$
The objective is to simplify the complex rational expression \(\frac{\frac{x^2}{x-1}}{\frac{3x^2}{x+4}}\). A rational expression is essentially a fraction where the numerator and denominator are polynomials. To simplify this, we must perform division of these fractions by multiplying the main numerator by the reciprocal of the main denominator. We must also identify the values of \(x\) for which the expression is undefined: the denominators cannot be zero, so \(x-1 eq 0\) (\(x eq 1\)), \(x+4 eq 0\) (\(x eq -4\)), and the entire denominator fraction \(\frac{3x^2}{x+4}\) cannot be zero, which means \(3x^2 eq 0\) (\(x eq 0\)).
To divide by a fraction, we multiply by its reciprocal. The expression is given by:
\[ \frac{\frac{x^2}{x-1}}{\frac{3x^2}{x+4}} = \frac{x^2}{x-1} \cdot \frac{x+4}{3x^2} \]
Multiplying these two fractions yields:
\[ \frac{x^2(x+4)}{(x-1)(3x^2)} \]
In the combined fraction \(\frac{x^2(x+4)}{3x^2(x-1)}\), we observe a common factor of \(x^2\) in both the numerator and the denominator. Provided that \(x
eq 0\), we can cancel this common term:
\[ \frac{x^2(x+4)}{3x^2(x-1)} = \frac{x+4}{3(x-1)} \]
This simplified expression is valid for all \(x
eq 0, 1, -4\). Thus, the fully simplified form of the original expression is \(\frac{x+4}{3(x-1)}\).
Simplifying rational expressions is a fundamental skill in algebra with many practical applications, such as in physics and engineering. For instance, when analyzing the velocity of an object that changes speed, you might encounter complex formulas representing ratios of motion. By simplifying these expressions, you can more easily solve for specific variables, like time or acceleration, making complex physical systems much easier to understand and predict. Understanding how to reduce these fractions ensures that you can handle more advanced modeling in fields like fluid dynamics or electrical circuit design.