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

Contents

Examples

Behaviour under addition, subtraction and multiplication

The degree of the zero polynomial

The degree computed from the function values

Extension to polynomials with two or more variables

Degree function in abstract algebra

Names of polynomials by degree

See also

References

Examples

★ The polynomial $3\; -\; 5\; x\; +\; 2\; x^5\; -\; 7\; x^9$ has degree 9.

★ The polynomial $(y\; -\; 3)(2y\; +\; 6)(-4y\; -\; 21)$ 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 $(y\; -\; 3)(2y\; +\; 6)(-4y\; -\; 21)$, 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, $z^5\; +\; 8\; z^4\; +\; 2\; z^3\; -\; 4\; z^2\; +\; 14\; z\; +\; 6$.

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
:$deg(PQ)\; =\; deg(P)\; +\; deg(Q)$.
For example:

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

The degree of the zero polynomial

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
:$max(a,-infty)\; =\; a$,
and
:$a\; +\; -infty\; =\; -infty$.
For example:

★ The degree of the sum $(x^3+x)+(0)=x^3+x$ is 3. Note that $3\; le\; max(3,\; -infty)$.

★ The degree of the difference $x-x\; =\; 0$ is $-infty$. Note that $-infty\; le\; max(1,1)$.

★ The degree of the product $(0)(x^2+1)=0$ is $(-infty)+2\; =\; -infty$.
The price to be paid for saving the rules for computing the degree of sums and products of polynomials is that the general rule
:$a+b=a\; quad\; Rightarrow\; quad\; b=0$,
breaks down when $a\; =\; -infty$.

The degree computed from the function values

If a polynomial ''f''(''x'') has positive values for sufficiently large values of ''x'', then the degree of that polynomial can be computed by the formula
:$deg(f)\; =\; lim\_\{x$
arrinfty}rac{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 ∞.

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 ''x''²''y''² + 3''x''³ + 4''y'' has degree 4, the same degree as the term ''x''²''y''².
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''.
:''x''²''y''² + 3''x''³ + 4''y'' = (3)''x''³ + (''y''²)''x''² + (4''y'') = (''x''²)''y''² + (4)''y'' + (3''x''³)
This polynomial has degree 3 in ''x'' and degree 2 in ''y''.

Degree function in abstract algebra

Given a ring R, the polynomial ring R[''x''] is the set of all polynomials in ''x'' that have coefficients chosen from R. In the special case that R is also a field, then the polynomial ring R[''x''] is a principal ideal domain and, more importantly to our discussion here, a euclidean domain.
It can be shown that the degree of a polynomial over a field satisfies all of the requirements of the ''norm'' function in the euclidean domain. That is, given two polynomials ''f''(''x'') and ''g''(''x''), the degree of the product ''f''(''x'')•''g''(''x'') must be larger than both the degrees of ''f'' and ''g'' individually. In fact, something stronger holds:
: deg( ''f''(''x'') • ''g''(''x'') ) = deg(''f''(''x'')) + deg(''g''(''x''))
For an example of why the degree function may fail over a ring that is not a field, take the following example. Let R = $mathbb\{Z\}/4mathbb\{Z\}$, the ring of integers modulo 4. This ring is not a field (and is not even an integral domain) because 2•2 = 4 (mod 4) = 0. Therefore, let ''f''(''x'') = ''g''(''x'') = 2''x'' + 1. Then, ''f''(''x'')•''g''(''x'') = 4''x''² + 4''x'' + 1 = 1. Thus deg(''f''•''g'') = 0 which is not greater than the degrees of ''f'' and ''g'' (which each had degree 1).
Since the ''norm'' function is not defined for the zero element of the ring, we consider the degree of the polynomial ''f''(''x'') = 0 to also be undefined so that it follows the rules of a norm in a euclidean domain.

Names of polynomials by degree

Polynomials with small degrees may be named according to their degree as follows:

★ Degree 1 - linear

★ Degree 2 - quadratic

★ Degree 3 - cubic

★ Degree 4 - quartic

★ Degree 5 - quintic

★ Degree 6 - sextic ''or'' hexic

★ Degree 7 - septic ''or'' heptic

★ Degree 8 - octic

★ Degree 9 - nonic

★ Degree 10 - decic

★ Degree 100 - hectic^[1]
The names beyond "quintic" are uncommon and rarely used.

See also

★ Degree (mathematics) — other meanings of ''degree'' in mathematics

References

1. Miami Dade College; ''Polynomials''