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

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The discussion revolves around determining the degree of the polynomial equation $xy + yz + xz + z^2x = y^4$ for various combinations of the variables x, y, and z. The correct degrees for each variable are identified as x being 1, y being 4, and z being 2, but there is confusion regarding the degrees of combinations like xz, yz, and xyz. It is clarified that the degree of a polynomial is defined by the highest power of the variable in each term. When considering combinations, treating other variables as constants helps in determining the degree, leading to the conclusion that the degrees for xz, yz, and xyz are 3, 4, and 4, respectively. Understanding these definitions and methods is crucial for accurately assessing polynomial degrees.
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$xy+yz+xz+z^2x=y^4$ 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!
 
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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".)
 
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!

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

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.
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.
 
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