How do you use the binomial theorem to expand and simplify the expression #3(x+1)^5-4(x+1)^3#?
The result is
Details of the expansions follow...
The first binomial will expand into
The trick is coming up with
Either way, the coefficients turn out to be 1, 5, 10, 10, 5, 1 in that order.
Now multiply the whole thing by 3.
Subtract from this the second expanded binomial (also get the coefficients in one of the ways mentioned above):
The finished result is found by combining like terms (those in #x^3, x^2, x, etc.)
Hope that all makes sense!