method of detached coefficients is simply a way of dealing with
algebra by dropping the algebraic variables and simply using the
numbers, keeping them in their correct places. Some examples follow.
Binomial Coefficients with a Calculator
Method of Detached Coefficients involves detaching the coefficients of
polynomial and dealing with the coefficients only. For instance: (1+x)2=1+2x+x2 112=121
(1+x)4=1+4x+6x2+4x3 +x4 114=14641
is, we can drop the x's, y's etc, and use the coefficients alone. In
the above cases, we find the coeffients simply by using 11 instead of
(1+x), and using arithmtic to expand our binomial (or whatever).
After 114, the calculator does not give us the clear result we seek:
This reminds us that we need to keep our numbers separate: 1015=10510100501
means we soon exceed the capacity of the ten-digit calculator, and need
to use Windows Calculator. However, adding a zero, keeps the
coefficients separate, so we can read of the results, and write: 1015=10510100501, as (a+b)5=a5+5a4b+10a3b2+10a2b3+5ab4+b5 However, an ordinary calculator gives: 1015=105101005 from which we can guess the last digit is 1.
The important thing is to keep the coefficients separate. A final example: