~Very useful for raising a 2-term expression
(binomial) to an positive integer power. It can also be used for any type of
power with an expanded interpretation of combinations.
~Examples: Use for (x+h)3, (2x-5)4 (be careful with a
negative, it belongs to the 2nd term in the formula).
~The formula: (a+b)n = an +nC1 a(n-1)b + nC2 a(n-2)b2 +
nC3 a(n-3)b3 + & so
on.
~Note: An easy way to remember is to look at the pattern. The
powers of a (the first term) start at n and go down one in each term proceeding
until it disappears in the last term. The power on b starts in the 2nd term with
a 1 and increases one in each term proceeding until it reaches the highest power
(n). The combinations start in the 2nd term ( n taken one at a time, then 2 at a time,
then 3 at a time, & so on until it reaches n at a time, which has a value of
1 at the end).
~When n is a positive whole number, the expansion terminates.
The last term will be bn.
~If n is relatively small, you can get these from Pascal's
Triangle (see link).
~If you need to express the combinations in terms of n, use the
formula nCr
= n!/r!(n-r)!
~If your binomial has a negative sign in the middle, this negative
belongs to b, i.e., for (x-2)5,
a=x & b= -2 in the expansion.
~Basic examples:
a) Expand (x+2)4. We get, x4 + 4C1 x3 (2) + 4C2 x2 (22) + 4C3 x (23) + 24,
or
simplifying, x4 + 8x3+24x2 +32x +
16
b) Expand (x+h)3. We get, x3 +
3x2(h) + 3xh2 + h3