Degree of a polynomial

The degree of a polynomial is the highest degree of its terms when the polynomial is expressed in its canonical form consisting of a linear combination of monomials. The degree of a term is the sum of the exponents of the variables that appear in it. The term order has been used as a synonym of degree but, nowadays, may refer to several other concepts (see order of a polynomial). For example, the polynomial $7x^2y^3 + 4x - 9$ has three terms. (Notice, this polynomial can also be expressed as $7x^2y^3 + 4x^1y^0 - 9x^0y^0$.) The first term has a degree of 5 (the sum of the powers 2 and 3), the second term has a degree of 1, and the last term has a degree of 0. Therefore, the polynomial has a degree of 5 which is the highest degree of any term.

To determine the degree of a polynomial that is not in standard form (for example $(x+1)^2-(x-1)^2$), one has to put it first in standard form by expanding the products (by distributivity) and combining the like terms; for example $(x+1)^2-(x-1)^2= 4x$ is of degree 1, even though each summand has degree 2. However, this is not needed when the polynomial is expressed as a product of polynomials in standard form, because the degree of a product is the sum of the degrees of the factors.

Names of polynomials by degree

The following names are assigned to polynomials according to their degree:^[1][2][3]

• Special case – zero (see § Degree of the zero polynomial below)
• Degree 0 – constant^[4]
• Degree 1 – linear
• Degree 2 – quadratic
• Degree 3 – cubic
• Degree 4 – quartic (or, if all terms have even degree, biquadratic)
• Degree 5 – quintic
• Degree 6 – sextic (or, less commonly, hexic)
• Degree 7 – septic (or, less commonly, heptic)

For higher degrees, names have sometimes been proposed,^[5] but they are rarely used:

• Degree 8 – octic
• Degree 9 – nonic
• Degree 10 – decic

Names for degree above three are based on Latin ordinal numbers, and end in -ic. This should be distinguished from the names used for the number of variables, the arity, which are based on Latin distributive numbers, and end in -ary. For example, a degree two polynomial in two variables, such as $x^2 + xy + y^2$, is called a "binary quadratic": binary due to two variables, quadratic due to degree two.^{[lower-alpha 1]} There are also names for the number of terms, which are also based on Latin distributive numbers, ending in -nomial; the common ones are monomial, binomial, and (less commonly) trinomial; thus $x^2 + y^2$ is a "binary quadratic binomial".

Other examples

• The polynomial $3 - 5 x + 2 x^5 - 7 x^9$ is a nonic polynomial
• The polynomial $(y - 3)(2y + 6)(-4y - 21)$ is a cubic polynomial
• The polynomial $(3 z^8 + z^5 - 4 z^2 + 6) + (-3 z^8 + 8 z^4 + 2 z^3 + 14 z)$ is a quintic polynomial (as the $z^8$ are cancelled out)

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, $z^5 + 8 z^4 + 2 z^3 - 4 z^2 + 14 z + 6$.

Behavior under polynomial operations

Behaviour under addition

The degree of the sum (or difference) of two polynomials is equal to or less than the greater of their degrees; the equality holds always when the degrees of the polynomials are different i.e.

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

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

E.g.

• 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)

Behaviour under scalar multiplication

The degree of the product of a polynomial by a non-zero scalar is equal to the degree of the polynomial, i.e.

$\deg(cP)=\deg(P)$.

E.g.

• The degree of $2(x^2+3x-2)=2x^2+6x-4$ is 2, just as the degree of $x^2+3x-2$.

Note that for polynomials over a ring containing divisors of zero, this is not necessarily true. For example, in $\mathbf{Z}/4\mathbf{Z}$, $\deg(1+2x) = 1$, but $\deg(2(1+2x)) = \deg(2+4x) =\deg(2) = 0$.

The set of polynomials with coefficients from a given field F and degree smaller than or equal to a given number n thus forms a vector space. (Note, however, that this set is not a ring, as it is not closed under multiplication, as is seen below.)

Behaviour under multiplication

The degree of the product of two polynomials over a field or an integral domain is the sum of their degrees

$\deg(PQ) = \deg(P) + \deg(Q)$.

E.g.

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

Note that for polynomials over an arbitrary ring, this is not necessarily true. For example, in $\mathbf{Z}/4\mathbf{Z}$, $\deg(2x) + \deg(1+2x) = 1 + 1 = 2$, but $\deg(2x(1+2x)) = \deg(2x) = 1$.

Behaviour under composition

The degree of the composition of two non-constant polynomials $P$ and $Q$ over a field or integral domain is the product of their degrees:

$\deg(P \circ Q) = \deg(P)\deg(Q)$.

E.g.

• If $P = (x^3+x)$, $Q = (x^2+1)$, then $P \circ Q = P \circ (x^2+1) = (x^2+1)^3+(x^2+1) = x^6+3x^4+4x^2+2$, which has degree 6.

Note that for polynomials over an arbitrary ring, this is not necessarily true. For example, in $\mathbf{Z}/4\mathbf{Z}$, $\deg(2x) \deg(1+2x) = 1\cdot 1 = 1$, but $\deg(2x\circ(1+2x)) = \deg(2+4x)=\deg(2) = 0$.

Degree of the zero polynomial

The degree of the zero polynomial is either left undefined, or is defined to be negative (usually −1 or −∞).^[6]

Like any constant value, the value 0 can be considered as a (constant) polynomial, called the zero polynomial. It has no nonzero terms, and so, strictly speaking, it has no degree either. As such, its degree is undefined. The propositions for the degree of sums and products of polynomials in the above section do not apply if any of the polynomials involved is the zero polynomial.^[7]

It is convenient, however, to define the degree of the zero polynomial to be negative infinity, −∞, and introduce the arithmetic rules^[8]

$\max(a,-\infty) = a ,$

and

$a + -\infty = -\infty .$

These examples illustrate how this extension satisfies the behavior rules above:

• The degree of the sum $(x^3+x)+(0)=x^3+x$ is 3. This satisfies the expected behavior, which is that $3 \le \max(3, -\infty)$.
• The degree of the difference $(x)-(x) = 0$ is $-\infty$. This satisfies the expected behavior, which is that $-\infty \le \max(1,1)$.
• The degree of the product $(0)(x^2+1)=0$ is $-\infty$. This satisfies the expected behavior, which is that $-\infty = -\infty + 2$.

Computed from the function values

The degree of a polynomial f 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 ∞.

Another formula to compute the degree of f from its values is

$\deg f = \lim_{x\to\infty}\frac{x f'(x)}{f(x)}.$

(This follows from L'Hôpital's rule.)

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 (sometimes called the total 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.

x²y² + 3x³ + 4y = (3)x³ + (y²)x² + (4y) = (x²)y² + (4)y + (3x³)

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 = $\mathbb{Z}/4\mathbb{Z}$, the ring of integers modulo 4. This ring is not a field (and is not even an integral domain) because 2 × 2 = 4 ≡ 0 (mod 4). Therefore, let f(x) = g(x) = 2x + 1. Then, f(x)g(x) = 4x² + 4x + 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.

Notes

1. For simplicity, this is a homogeneous polynomial, with equal degree in both variables separately.

1. "Names of Polynomials". November 25, 1997. Retrieved 5 February 2012.
2. Mac Lane and Birkhoff (1999) define "linear", "quadratic", "cubic", "quartic", and "quintic". (p. 107)
3. King (2009) defines "quadratic", "cubic", "quartic", "quintic", "sextic", "septic", and "octic".
4. Shafarevich (2003) says of a polynomial of degree zero, $f(x)=a_0$: "Such a polynomial is called a constant because if we substitute different values of x in it, we always obtain the same value $a_0$." (p. 23)
5. James Cockle proposed the names "sexic", "septic", "octic", "nonic", and "decic" in 1851. (Mechanics Magazine, Vol. LV, p. 171)
6. Shafarevich (2003) says of the zero polynomial: "In this case, we consider that the degree of the polynomial is undefined." (p. 27)
   Childs (1995) uses −1. (p. 233)
   Childs (2009) uses −∞ (p. 287), however he excludes zero polynomials in his Proposition 1 (p. 288) and then explains that the proposition holds for zero polynomials "with the reasonable assumption that $-\infty$ + m = $-\infty$ for m any integer or m = $-\infty$".
   Axler (1997) uses −∞. (p. 64)
   Grillet (2007) says: "The degree of the zero polynomial 0 is sometimes left undefined or is variously defined as −1 ∈ ℤ or as $-\infty$, as long as deg 0 < deg A for all A ≠ 0." (A is a polynomial.) However, he excludes zero polynomials in his Proposition 5.3. (p. 121)
7. Barile, Margherita, "Zero Polynomial", MathWorld.
8. Axler (1997) gives these rules and says: "The 0 polynomial is declared to have degree $-\infty$ so that exceptions are not needed for various reasonable results." (p. 64)

References

• Axler, Sheldon (1997), Linear Algebra Done Right (2nd ed.), Springer Science & Business Media
• Childs, Lindsay N. (1995), A Concrete Introduction to Higher Algebra (2nd ed.), Springer Science & Business Media
• Childs, Lindsay N. (2009), A Concrete Introduction to Higher Algebra (3rd ed.), Springer Science & Business Media
• Grillet, Pierre Antoine (2007), Abstract Algebra (2nd ed.), Springer Science & Business Media
• King, R. Bruce (2009), Beyond the Quartic Equation, Springer Science & Business Media
• Mac Lane, Saunders; Birkhoff, Garrett (1999), Algebra (3rd ed.), American Mathematical Society
• Shafarevich, Igor R. (2003), Discourses on Algebra, Springer Science & Business Media

External links

• Polynomial Order; Wolfram MathWorld