That's because we want the degree of the product of two polynomials to be the sum of the degrees of the two polynomials, and if 0 had a defined degree, the:
I'm not sure what this question means. Numbers do not have " degrees " in any normal sense of the word.
If you mean in the sense that numbers can represent the square, the cube, or any other power of some other number, then any number can be considered to have any number of " degrees " - for instance, 256 is the square of 16, the fourth power of 4, the eighth power of 2, and so on, so could be considered to be of " degree " 2, or 4, or 8, according to circumstances ( of course it is also the cube of 256^(1/3), or the fifth power of 256^(1/5), and so on).
Zero, however, is a special case in many ways, and is excluded from many of the operations which are applied to the other numbers, or the operations have to be defined in a particular way when applied to zero - the most common example of this is division by zero, which is "undefined", since there is no sensible answer to such an operation within the number system.
So you could say that 0 is of no degree, or of any degree, depending on how you choose to define it - assuming that you had some purpose in defining it, and that it made sense in some system of logic !