Пример 4. Сократите дробь: \(\frac{p(a)}{p(\frac{1}{a})}\), если \(p(x)=(x+\frac{6}{x})(6x+\frac{1}{x})\).

Решение:

1. Найдем \(p(a)\): \(p(a) = (a+\frac{6}{a})(6a+\frac{1}{a}) = (\frac{a^2+6}{a})(\frac{6a^2+1}{a}) = \frac{(a^2+6)(6a^2+1)}{a^2}\).
2. Найдем \(p(\frac{1}{a})\): \(p(\frac{1}{a}) = (\frac{1}{a}+\frac{6}{\frac{1}{a}})(6\frac{1}{a}+\frac{1}{\frac{1}{a}}) = (\frac{1}{a}+6a)(\frac{6}{a}+a) = (\frac{1+6a^2}{a})(\frac{6+a^2}{a}) = \frac{(6a^2+1)(a^2+6)}{a^2}\).
3. Сократим дробь:

\(\frac{p(a)}{p(\frac{1}{a})} = \frac{\frac{(a^2+6)(6a^2+1)}{a^2}}{\frac{(6a^2+1)(a^2+6)}{a^2}} = 1\)

Ответ: 1