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Topic: What is a binomial?

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Subject:   What is a binomial?
Author: Eric Goolish
Date: Apr 29 2005
Recently, I had a 60 minute discussion with fellow high school teachers about
what is a binomial?

The resources I have suggest a binomial is a polynomial, therefore it is the sum
of monomials, specially two monomials.  However, there is no discussion of
simplified form.

So playing devil's advocate, is 5 + 2 a binomial?  I don't think so because it
can be simplifed to 7, which is a monomial.

However, one textbook I use for class suggests that (a sqrt b) + (c sqrt d) is a
binomial.  Specifically, they say BINOMIALS of the form (a sqrt b) + (c sqrt d)
and (a sqrt b) - (c sqrt d) are conjugates.

Personally, I feel these can be called binomials because there are two terms
when simplied.  For instance, sqrt 2 + sqrt 3 (Binomial).

However, it can also be suggested that sqrt 2 + sqrt 3 is just one entity, one
number, so is it a monomial?  Perhaps, (sqrt 2 + sqrt 3)x^0 - representing the
degree of 0, thus a constant, thus a monomial?

What does everyone think about this?  The more I dig into these math ideas, the
more I think there is much confussion in our notation and terminology.  How can
1 + 1/2 be written as 1 1/2 (one and a half) with no addition sign, but now if
we have 2 + sqrt 3, we cannot put them together and write 2 sqrt 3 because that
would mean multiplication.  I think there is much confusion here and we should
not be surprised when students 'don't get it.'

What does everyone think???

Eric Goolish (Curie Metro High Student Teacher)

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