Look at the term 3x^2 y. Two different letters — x and y — stand side by side. A natural question jumps out: is this one polynomial or two? And when a problem asks for "the degree," which degree does it mean — the exponent on x, the exponent on y, or the sum?
3x^2 y is one polynomial in two variables. And the word "degree" splits into three meanings when more than one variable is present. This article sorts out the vocabulary.
How many variables
Count the distinct letters. In 3x^2 y, the letters are x and y, so this is a polynomial in 2 variables. Compare with 3x^2, which has only x — one variable. And 2xyz, which has x, y, and z — three variables.
Three kinds of degree for multivariable polynomials
Once more than one variable is in play, the single word "degree" is not enough. Three distinct quantities all go by that name.
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Total degree (overall degree). For a single term, this is the sum of the exponents on all the variables. For 3x^2 y, the sum is 2 + 1 = 3. For a polynomial with multiple terms, the total degree is the largest such sum across its terms.
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Degree in x (or any one chosen variable). Treat every other variable as a constant, and look at the highest exponent on x. For 3x^2 y, the degree in x is 2.
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Degree in y. Treat x as constant, read off the highest exponent on y. For 3x^2 y, the degree in y is 1.
So 3x^2 y has three different "degrees" — 3 overall, 2 in x, 1 in y — and no single number is the degree. You have to say which one you mean.
Worked examples
- 5x^2 y^3 — one term. Total degree 2 + 3 = 5. Degree in x is 2; in y, 3.
- 2x^2 + 3xy^2 — total degrees of the two terms are 2 and 3; overall total 3. Degree in x: the exponents are 2 and 1, so 2. Degree in y: 0 and 2, so 2.
- x^5 + y^7 — overall total 7. Degree in x is 5; in y, 7.
- x^2 + xy + y^2 — each term has total degree 2; overall total 2. Degree in x is 2; in y, 2.
Notice that in the last example every term has the same total degree — that structure has a name.
Which "degree" matters depends on the problem
The three degrees each answer a different kind of question.
- Simplification and classification. When a textbook says "degree" with no qualifier, it means the total degree. Total degree 2 is "quadratic," total degree 3 is "cubic," and so on.
- Factoring with respect to a variable. If you factor x^2 + 2xy + y^2 by "treating it as a polynomial in x," what matters is the degree in x (here 2) — it tells you how many linear factors in x to look for.
- Substitution. If you fix a specific value for y and solve for x, the degree in x controls how hard the resulting one-variable equation is.
- Geometry of the zero set. When a polynomial equation in two variables is drawn as a curve (like x^2 + y^2 = r^2), the total degree controls how the curve meets lines. Bézout's theorem says a curve of total degree d meets a line in at most d points.
Homogeneous polynomials
A polynomial is homogeneous of degree d when every term has total degree exactly d.
- x^2 + xy + y^2 — each term has total degree 2. Homogeneous of degree 2.
- x^2 + y^2 — homogeneous of degree 2.
- x^3 + x^2 y + xy^2 + y^3 — homogeneous of degree 3.
- x^2 + x + 1 — degrees 2, 1, 0 across terms. Not homogeneous.
Homogeneous polynomials have nicely behaved zero sets: scaling any solution (x, y) by t gives another solution (tx, ty), so the zero set is a union of lines through the origin.
The case of constants
A plain number like 5 has:
- Total degree 0. Think of 5 = 5x^0 y^0.
- Degree in any particular variable 0. The variable does not appear.
A nonzero constant has degree 0, not undefined. The one exception is the zero polynomial (every coefficient is zero): its degree is left undefined or written as -\infty, by convention.
Three or more variables
Nothing new happens; the rules extend directly. For 2xyz: three variables, each with exponent 1, so total degree 3 and degree 1 in each variable. For 4x^2 y z^3: total degree 6; degree in x is 2, in y is 1, in z is 3.
Treating a multivariable polynomial as single-variable
Sometimes one variable is "privileged" — it is the unknown you want to solve for, while the other is a parameter. Then it pays to view the polynomial as a single-variable one whose coefficients involve the other letter.
Take x^2 + yx + y^2. As a polynomial in x, its coefficients are a_2 = 1, a_1 = y, a_0 = y^2. Now every tool for quadratics applies, with y tagging along as a parameter.
Example. For which y does x^2 + 2xy + y^2 = 9 have a real x-solution? Rewrite as x^2 + 2y \cdot x + (y^2 - 9) = 0. The discriminant is
positive for every y, so the equation has a real x-solution for every real y.
In physics and geometry
Multivariable polynomials appear whenever a shape is written in coordinates. x^2 + y^2 = r^2 is a circle (total degree 2). x^2 + y^2 + z^2 = r^2 is a sphere (three variables, total degree 2). y^2 = 4ax is a parabola. x^3 + y^3 = 3xy is the folium of Descartes (total degree 3). The shape of the zero set depends on both the total degree and the number of variables.
Recognition drill
State the total degree and the degree in each variable:
- 3x^2 y: total 3; in x, 2; in y, 1.
- x^3 y^2 + xy^3: total 5. In x: 3. In y: 3.
- 4x + 5y - 7: total 1. In x: 1; in y: 1.
- x^2 y^2 + x^2 z^2: total 4. In x: 2; in y: 2; in z: 2.
- xy + yz + xz: total 2. In x: 1; in y: 1; in z: 1.
Recipe: for total degree, sum exponents within each term and take the largest such sum; for degree in a specific variable, look at every term and take the largest exponent on that variable.
Common confusions
- "3xy is linear." It is linear in x alone and linear in y alone, but its total degree is 2. An expression like 3xy is called bilinear — linear in each variable separately, but quadratic overall.
- "Two variables means two polynomials." No. 3x^2 y is one polynomial in the ring \mathbb{R}[x, y], not a pair. The variables are tangled together by multiplication inside a single expression.
- "The degree of a constant is undefined." Only the zero polynomial has undefined degree. A nonzero constant like 5 has degree 0.
Closing
Multivariable polynomials carry three flavours of degree — total degree, and one per-variable degree for each variable. Total degree is the default whenever a problem says "degree." Per-variable degrees come in when you solve, factor, or slice with respect to one variable.
For the opener: 3x^2 y is one polynomial in two variables, with total degree 3, degree 2 in x, and degree 1 in y.