MHB Determine the degree of each equation in each of the indicated unknowns.

[paulmdrdo1]

$xy+yz+xz+z^2x=y^4$ 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!

 

[HallsofIvy]

You (and perhaps the person who set this question) seem to have the wrong idea about "degree". The "degree" of a polynomial equation, in each variable, is the highest power to which the variable appears. You say, "I know x is 2nd degree" but I see no "x^2" in the given polynomial. Perhaps that was a mistype. But I have no idea what could be meant by the degree of "x and z", "y and z", or "x, y, and z".

(If you had said "degree of xz" then I might stretch a point and seeing that term "xz" and "z^2x= z(xz)" say that the polynomial is of degree one in "xz".)

 

[chisigma]

$xy+yz+xz+z^2x=y^4$ 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!

It seems the degree of x to be 1... or may be the expression is erroneous?... Kind regards $\chi$ $\sigma$

 

[paulmdrdo1]

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.

 

[Evgeny.Makarov]

First, let's agree on the basic definitions. From Wikipedia:

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.

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.

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.