The
method of detatched 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
The
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)3=1+3x+3x2+x3 113=1331
(1+x)4=1+4x+6x2+4x3 +x4 114=14641
That
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:
115=161051
This reminds us that we need to keep our numbers separate: 1015=10510100501
This
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:
We cannot so easily use the calculator in the case of multinomial coefficients of the form:(a+b+c)2 (a+b+c)2=a2+b2+c2+2ab+2ac+2bc
We cannot do this because if we write it as 111, for instance, with a=100, b=10, and c=1, we get confused between b2=100 and ac=100, and 1112=12321 with the 3 being a confusion between b2 and 2ac.
To draw out the difference we can write: 100010012=100020021002001 where a=10,000,000, b=1000, and c=1.
The first 1 is a2, the first 2 is 2ab, the next 2ac. The second 1 is b2 and the final 2 is 2bc, and the final 1 c2. (a+b+c)2=a2+b2+c2+2ab+2ac+2bc
Another example: 100010013=1000300330061033003001
This implies that (a+b+c)3=a3+b3+c3+3a2b+3a2c+3ab2+3ac2+3b2c+3bc2+6abc
Multinomials of the form (1+x+x2)n
Multinomials of the form (1+x+x2) have a basic order, so: 1112=12321
