Question

Show that (3a + 2b-c+d)^2 - 12a(2b-c+d) is a perfect square.
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Answer 1

Sum of Coefficients

If we make x and y equal to 1 in the following (Binomial Expansion)

multinomialExpansion1.gif [1.1]

We find the sum of the coefficients:

multinomialExpansion2.gif [1.2]

Another way to look at 1.1 is that we can select an item in 2 ways (an x or a y), and as there are n factors, we have, in all, 2n possibilities.

Sum of Coefficients for p Items

Where there are p items:

multinomialExpansion3.gif [1.3]

We can set each of the x's to 1 so the value depends only on the sum of the coefficients, so the sum of the coefficients is pn.

In another sense, we can choose one of the items in p ways from the n factors, obtaining pn different ways to select the terms of the series.

Squaring The Multinomial

Converting to Binomial

We can square a multinomial using a generalisation of the binomial theorem multi1.gif [2.1]

For instance, below, A=a+b, and B=c+d:

multi2.gif [2.1]

Giving:

multi3.gif [2.2]

Or:

multi4.gif [2.3]

Combination Approach

Here we "think out" the terms using our knowledge of combinations.