Degree of a polynomial

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The degree of a polynomial in one variable is the largest power with which the variable appears with non-zero coefficient. Thus, for instance, the degree of $P_1(x) = 2x^3 -7x^2 + 100$ is 3 and the degree of $P_n(x) = \sum_{i = 0}^n x^i = 1 + x + x^2 + \ldots + x^n$ is $n$.

In general, the degree of linear polynomials is 1 and the degree of constant functions is 0, except that sometimes the degree of the zero polynomial $P(x) = 0$ is taken to be $-\infty$ by convention.

For polynomials in several variables, one typically computes the degree by adding the exponents of the variables in each term and taking the largest of these sums. For example, the polynomial $Q(x, y, z) = x^2yz + 2xz^3 + y^4z^2$ is taken to be the largest of $2 + 1 + 1, 1 +0+ 3$ and $0 + 4 + 2$, and so is 6. Note that this is not in general the same as the degree of the polynomial that arises from setting all variables equal; for instance, $R(x, y, z) = x^2yz - 2xy -z^4$ is degree 4 but $R(x, x, x) = -2x^2$ is only degree 2.

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