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Suggestions about exercise 10 in Sec 2.6

*Updated October 29, 2008*

*1.* First, a basic definition: saying that two monomials
are *relatively prime* means that they have no common factor.

For instance, in 3 variables, *xy* and *xz*
are *not* relatively prime because *y* is a common factor.
On the other hand,

*xy* and *z*² are
relatively prime.

*2.* Now a question: if two monomials are relatively prime,
what is their LCM? Similar terminology is used for integers.

For instance,
6 and 25 are relatively prime: what is their LCM? A very similar answer is
valid for relatively prime monomials.

*3.* The equation that you're asked to prove in part *a*
certainly is true, but something slightly different may be more useful in
part *b*.

First, note that *fg* is both added and
subtracted in the expression on the right side. Therefore, that equation
is equivalent to

the following equation:

*S*(*f,g*) = LT(*g*)*f* - LT(*f*)*g*

And remarkably, this equation is *both* more useful in
part *b* {in my opinion, anyway} *and also* easier
to prove.

{Just make appropriate substitutions from what is given, and the
answer to the question asked above,

and then do a bit of algebraic
calculation.}
Comments and questions to:
`roberts@math.umn.edu`

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