Degree of a polynomial

Degree of a polynomial

The degree of a polynomial represents the highest degree of a polynominal's terms (with non-zero coefficient), should the polynomial be expressed in canonical form (i.e. as a sum or difference of terms). The degree of an individual term is the sum of the exponents acting on the term's variables. The word degree has been favored for some decades in standard textbooks - but in some older books, the word order may be used instead.

For example, the polynomial 7x2y3 + 4x − 9 has three terms. (Notice, this polynomial can also be expressed as 7x2y3 + 4x1y0 − 9x0y0.) 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 (y − 3)(2y + 6)( − 4y − 21)) it is easier to expand or express the polynomial into a sum or difference of terms; this may be achieved by multiplying each of its factors, and combining monomial terms. This makes the exponents more obvious, and easier to determine when calculating the degree of the equation. Since, (y − 3)(2y + 6)( − 4y − 21) = − 8y3 − 42y2 + 72y + 378, the degree of the polynomial can be found to be 3.

Names of polynomials by degree

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

• Degree 0 – constant
• Degree 1 – linear
• Degree 3 – cubic
• Degree 4 – quartic (or, less commonly, biquadratic) (or, a little more common, Fourth degree)
• Degree 5 – quintic
• Degree 6 – sextic (or, less commonly, hexic)
• Degree 7 – septic (or, less commonly, heptic)
• Degree 8 – octic
• Degree 9 – nonic
• Degree 10 – decic
• Degree 100 - hectic

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

Other examples

• The polynomial 3 − 5x + 2x5 − 7x9 has degree 9.
• The polynomial (y − 3)(2y + 6)( − 4y − 21) has degree 3.
• The polynomial (3z8 + z5 − 4z2 + 6) + ( − 3z8 + 8z4 + 2z3 + 14z) has degree 5.

The canonical forms of the three examples above are:

• for 3 − 5x + 2x5 − 7x9, after reordering, − 7x9 + 2x5 − 5x + 3;
• for (y − 3)(2y + 6)( − 4y − 21), after multiplying out and collecting terms of the same degree, − 8y3 − 42y2 + 72y + 378;
• for (3z8 + z5 − 4z2 + 6) + ( − 3z8 + 8z4 + 2z3 + 14z), in which the two terms of degree 8 cancel, z5 + 8z4 + 2z3 − 4z2 + 14z + 6.

Behavior under addition, subtraction, multiplication and function composition

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

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

e.g.

• The degree of (x3 + x) + (x2 + 1) = x3 + x2 + x + 1 is 3. Note that 3 ≤ max(3, 2)
• The degree of (x3 + x) − (x3 + x2) = − x2 + 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).

e.g.

• The degree of (x3 + x)(x2 + 1) = x5 + 2x3 + x is 3 + 2 = 5.

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

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

e.g.

• If P = (x3 + x), Q = (x2 + 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.

The degree of the zero polynomial[citation needed]

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. 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 the degree of the zero polynomial to be 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

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.[citation needed]
• The degree of the exponential function, $\ \exp x$, is ∞.

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

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

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.

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 (mod 4) = 0. Therefore, let f(x) = g(x) = 2x + 1. Then, f(x)•g(x) = 4x2 + 4x + 1 = 1. Thus deg(fg) = 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.

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