1.6.1—Binomial expansion

For a positive integer n, (a+b)^n=Σᵣ C(n,r)a^{n−r}b^r. The general term identifies the power and coefficient without writing every term.

Track powers of the variable and signs carefully; for a coefficient question, set the target power before simplifying. The expansion is finite only for non-negative integer n.

The x² term in (1+2x)^5 is C(5,2)·(2x)²=40x².

The coefficient C(n,r) is not the whole term; the powers of both factors still contribute.