a binomial thm problem
$\displaystyle (1 + 2x)^4(1- x)^3$
The most clever thing I can come up with is to define two new variables:
$\displaystyle y = 1 + \frac{x}{2}$
and
$\displaystyle z = \frac{3x}{2}$
Then
$\displaystyle y + z = 1 + 2x$
and
$\displaystyle y - z = 1 - x$
Thus
$\displaystyle (1 + 2x)^4(1- x)^3 = (y + z)^4(y - z)^3$
$\displaystyle = [(y + z)^3(y - z)^3](y + z)$
$\displaystyle = (y^2 - z^2)^3(y + z)$
$\displaystyle = ([1 + 2x]^2 - [1 - x]^2)^3(1 + 2x)$
etc.
But I doubt this actually cuts down on the work load.
-Dan