The discussion revolves around determining the degree of a polynomial equation in various unknowns, specifically for the equation $xy+yz+xz+z^2x=y^4$. Participants explore how to assess the degree of individual variables as well as combinations of variables, including potential misunderstandings about the definitions of degree in polynomial contexts.
Participants express differing views on the degrees of the variables and combinations, with no consensus reached on the correct interpretation or calculation of these degrees.
There are unresolved assumptions regarding the definitions of degree in polynomial equations and how to apply these definitions to combinations of variables. The discussion reflects varying interpretations of the problem and the definitions involved.
paulmdrdo said:$xy+yz+xz+z^2x=y^4$ x;y;z; x and z; y and z; x, y, and z
I know x is 2nd degree, y is 4th degree, z is 2nd degree.
but I don't know how to determine the degree of the combinations of the unknowns. please help. thanks!
A polynomial can either be zero or can be written as the sum of a finite number of non-zero terms. Each term consists of the product of a number—called the coefficient of the term—and a finite number of indeterminates, raised to integer powers. The exponent on an indeterminate in a term is called the degree of that indeterminate in that term; the degree of the term is the sum of the degrees of the indeterminates in that term, and the degree of a polynomial is the largest degree of anyone term with nonzero coefficient.
My guess is that the degree of the equation in $xz$ is the degree of that equation where $y$ is considered a constant (i.e., a coefficient) rather than a variable (indeterminate). Then the degree of the right-hand size $y^4$ is 0, and the term with the highest degree is $z^2x$; its degree is 3.paulmdrdo said:what I mean is the degree of the polynomial equation in xz, the degree in yz, the degree in xyz. the answer in my book says 3,4,4 respectively. but I didn't understand why is that.