Find the Product:
\[ \left(\frac{3}{x} – 2x^2\right) \left(\frac{9}{x^2} + 4x^4 + 6x\right) \]
Solution:
Rearranging the terms:
\[ \left(\frac{3}{x} – 2x^2\right) \left(\frac{9}{x^2} + \frac{6x^3}{x^2} + 4x^4\right) \]
\[ = \left(\frac{3}{x} – 2x^2\right) \left(\frac{9}{x^2} + 6x + 4x^4\right) \]
Using identity:
\[ (a-b)(a^2+ab+b^2)=a^3-b^3 \]
Here, \[ a=\frac{3}{x},\qquad b=2x^2 \]
\[ \left(\frac{3}{x} – 2x^2\right) \left[\left(\frac{3}{x}\right)^2 + \left(\frac{3}{x}\right)(2x^2) + (2x^2)^2\right] \]
\[ = \left(\frac{3}{x}\right)^3 – (2x^2)^3 \]
\[ = \frac{27}{x^3} – 8x^6 \]