The easy way if your exponent is less than 10 or so, is to use Pascal's
Triangle to develop your coefficients, and then just fill in the pattern.
You need the first 6 rows of Pascal's Triangle to do a 5th degree expansion:
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
So if you had, the expansion would be:
Notice that the coefficients are just the last row of Pascal's Triangle, the
exponent on a starts at 5 and goes down by one per term until you get to the
last term where a disappears because the exponent is 0 (remember,)
and the exponent on b starts at 0 and increases by 1.
Now all you have to do is take the expansion above and replace a with 4x and b with -7
Notice that the 4 in 4x is inside the parentheses and must also be raised to the indicated power, so:
By the way, if you need to get the next or subsequent rows of Pascal's
Triangle, you start with 1 and then each number across the row is the sum of
the numbers immediately above to the right and left. The sixth row starts 1,
6, 15...
In general, the r-th coefficient of the n-th degree binomial expansion is