Degree of a polynomial

From Wikipedia, the free encyclopedia

The degree of a polynomial is the maximum of the degrees of all terms in the polynomial. For example, in 2x³ + 4x² + x + 7, the term of highest degree is 2x³; this term, and therefore the entire polynomial, are said to have degree 3. Sometimes the same concept is called the order of the polynomial.

1 Examples
2 Behaviour under addition, subtraction and multiplication
3 The degree of the zero polynomial is minus infinity
4 The degree computed from the function values
5 Extension to polynomials with two or more variables
6 See also

[edit] Examples

The polynomial $3 - 5 x + 2 x 5 - 7 x 9$ has degree 9.
The polynomial $(y - 3)(2 y + 6)( - 4 y - 21)$ has degree 3.
The polynomial $(3 z 8 + z 5 - 4 z 2 + 6) + ( - 3 z 8 + 8 z 4 + 2 z 3 + 14 z)$ has degree 5.

In general, to determine the degree of a polynomial expression, the expression has to be brought in "canonical form" by multiplying out until all terms are a product of constants and variables, in which terms with the same product of variables are collected together, and terms in which the constant factor is zero are elided. Usually (but not necessarily) the terms are also ordered from highest to lowest degree. The canonical forms of the three examples above are:

for $3 - 5 x + 2 x 5 - 7 x 9$ , after reordering, $- 7 x 9 + 2 x 5 - 5 x + 3$ ;
for $(y - 3)(2 y + 6)( - 4 y - 21)$ , after multiplying out and collecting terms of the same degree, $- 8 y 3 - 42 y 2 + 72 y + 378$ ;
for $(3 z 8 + z 5 - 4 z 2 + 6) + ( - 3 z 8 + 8 z 4 + 2 z 3 + 14 z)$ , in which the two terms of degree 8 cancel, $z 5 + 8 z 4 + 2 z 3 - 4 z 2 + 14 z + 6$ .

[edit] Behaviour under addition, subtraction and multiplication

The degree of the sum (or difference) of two polynomials is equal to or less than the greater of their degrees i.e.

$\deg(P + Q) \leq \max(\deg(P),\deg(Q))$ .

$\deg(P - Q) \leq \max(\deg(P),\deg(Q))$ .

For example:

The degree of $(x 3 + x) + (x 2 + 1) = x 3 + x 2 + x + 1$ is 3. Note that 3 ≤ max(3,2)
The degree of $(x 3 + x) - (x 3 + x 2) = - x 2 + x$ is 2. Note that 2 ≤ max(3,3)

The degree of the product of two polynomials is the sum of their degrees

deg(P Q) = deg(P) + deg(Q)

For example:

The degree of $(x 3 + x)(x 2 + 1) = x 5 + 2 x 3 + x$ is 3+2 = 5.

[edit] The degree of the zero polynomial is minus infinity

The function f(x)=0 is a polynomial, called the zero polynomial. It has no terms, and so, strictly speaking, it has no degree either. The above rules for the degree of sums and products of polynomials do not apply if any of the polynomials involved is the zero polynomial.

It is convenient, however, to define that the degree of the zero polynomial is minus infinity, −∞, and introduce the rules

max(a,−∞) = a,

and

a+(−∞) = −∞.

For example:

The degree of the sum $(x 3 + x) + (0) = x 3 + x$ is 3. Note that 3 ≤ max(3,−∞)
The degree of the difference x−x=0 is −∞. Note that −∞ ≤ max(1,1)
The degree of the product $(0)(x 2 + 1) = 0$ is (−∞)+2 = −∞.

The price to be paid for saving the rules for computing the degree of sums and products of polynomials is that the general rule

a+b=a implies that b=0,

breaks down when a=−∞.

[edit] The degree computed from the function values

If a polynomial f(x) has positive values for sufficient large values of x, then the degree of that polynomial can be computed by the formula

$\deg(f) = \lim_{x\rarr\infty}\frac{\log(f(x))}{\log(x)}$

This formula generalizes the concept of degree to some functions that are not polynomials. For example:

The degree of the multiplicative inverse, $\ 1/x$ , is −1.
The degree of the square root, $\sqrt x$ , is 1/2.
The degree of the logarithm, $\ \log x$ , is 0.
The degree of the exponential function, $\ \exp x$ , is ∞.

[edit] Extension to polynomials with two or more variables

For polynomials in two or more variables, the degree of a term is the sum of the exponents of the variables in the term; the degree of the polynomial is again the maximum of the degrees of all terms in the polynomial. For example, the polynomial x²y² + 3x³ + 4y has degree 4, the same degree as the term x²y².

However, a polynomial in variables x and y, is a polynomial in x with coefficients which are polynomials in y, and also a polynomial in y with coefficients which are polynomials in x.