Hallar el mínimo común denominador de $\frac{-7}{x^{2}-6 x-27}$ y $\frac{3}{x^{2}-49}$.
Answer
Thus, the minimum common denominator of \(-\frac{7}{x^{2}-6 x-27}\) and \(\frac{3}{x^{2}-49}\) is \(\boxed{x^{4} - 6x^{3} - 76x^{2} + 294x + 1323}\).
Step 1 ：Given two fractions, \(-\frac{7}{x^{2}-6 x-27}\) and \(\frac{3}{x^{2}-49}\), we are asked to find the minimum common denominator.
Step 2 ：First, we factorize the denominators of the two fractions. The expression \(x^{2}-6 x-27\) can be factorized into \((x-9)(x+3)\) and the expression \(x^{2}-49\) can be factorized into \((x-7)(x+7)\).
Step 3 ：Next, we find the least common multiple (LCM) of these two expressions. The LCM of \((x-9)(x+3)\) and \((x-7)(x+7)\) is \(x^{4} - 6x^{3} - 76x^{2} + 294x + 1323\).
Step 4 ：Thus, the minimum common denominator of \(-\frac{7}{x^{2}-6 x-27}\) and \(\frac{3}{x^{2}-49}\) is \(\boxed{x^{4} - 6x^{3} - 76x^{2} + 294x + 1323}\).
