Question

How to find the fourth term of a binomial expansion?

Answer 1

So let's look at how to expand a binomial..
Start with just the variable exponents which will be numbered in ascending and descending order.

example (without coefficients)
(x+y)^3 = x^3(y^0) + x^2(y^1) + x^1(y^2) + x^0(y^3)

If the binomial is (x-y)^3 the +/- signs will toggle like so.. (again leaving out coefficients)
(x-y)^3 = x^3(y^0) - x^2(y^1) + x^1(y^2) - x^0(y^3)

If there is a coefficient or power inside the binomial like (x^2+2y)^3 these are also raised to the appropriate power.

(x^2+2y)^3 = (x^2)^3(y^0)(2^0) + (x^2)^2(y^1)(2^1) + ....

now there are also coefficients for each term which follow a combination pattern of Pascal triangle. C(3,0) ; C(3,1) ; C(3,2) ; C(3,3)
these get multiplied by each term.

So the 4th term of (x^2-2y)^3
will be negative
have coefficient of: C(3,3) * (2^3) = 8
variables: x^0(y^3) = y^3
= -8y^3