I'm a baby boomer who was taught that the constant term of a polynomial is a coefficient, being the constant factor for the x^0 term.

That's not what's taught today.

Current text books are vague on their definition of coefficient, but, when they ask for a student to list all coefficient in a given polynomial, they do not include the constant term in their answer keys.

The Kahn Academy video on the topic does include the constant term as a coefficient, as does the related Wikipedia article.

However, all modern text books written to the Common Core that I have seen imply that the constant term is not a coefficient. A lot of web sites created by today's teachers who are teaching to the Common Core explicitly state that the constant term is not a coefficient.

If you have a polynomial of one variable, aren't the coefficients supposed to uniquely determine that polynomial? Aren't they all a computer program needs to know to work with polynomials? Without the constant term, you get the related polynomial that represents a vertical displacement of the original polynomial by the opposite of the constant term. But, you don't get the original polynomial.

Further, if we are talking about natural number constants, isn't each digit, including the ones' place, a coefficient of a power of 10? We don't segregate that digit by calling it something else; it's a digit similar to all the other digits.

Isn't mathematics supposed to be non-arbitrary and consistent?

I would like to see this taught as: after combining like terms, all constant factors, including the constant term, are coefficients. Two of the coefficients are special. The coefficient of the term with the largest exponent is the leading coefficient. The coefficient of the term with the smallest exponent (aka zero) is the constant term. That shouldn't be too much for eighth graders who were taught exponent properties, including the zero exponent property, in seventh grade.

Is this going to be yet another contributor to generational divide?