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Ken Ward's Mathematics Pages

Method of Detached Coefficients

Main Arithmetic Algebra Page

Page Contents

  1. Method of Detached Coefficients 
  2. Binomial Coefficients with a Calculator
  3. Multinomial Coefficients
    1. Multinomial Coefficients for (a+b+c)
    2. Multinomials of the form (1+x+x2)

Method of Detached Coefficients 

The 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

The Method of Detached Coefficients involves detaching the coefficients of polynomial and dealing with the coefficients only. For instance:


(1+x)4=1+4x+6x2+4x3 +x4

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:


This reminds us that we need to keep our numbers separate:

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
However, an ordinary calculator gives:
from which we can guess the last digit is 1.

The important thing is to keep the coefficients separate. A final example:

Implying correctly that:

Multinomial Coefficients

Multinomials of the form (a+b+c)n

We cannot so easily use the calculator in the case of multinomial coefficients of the form:(a+b+c)2

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
with the 3 being a confusion between b2 and 2ac.

To draw out the difference we can write:
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.

Another example:

This implies that

Multinomials of the form (1+x+x2)n

Multinomials of the form (1+x+x2) have a basic order, so:

Correctly implying that:

Ken Ward's Mathematics Pages

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