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| {{Use dmy dates|date=July 2013}}
| | I am Oscar and I totally dig that title. To gather badges is what her family and her enjoy. He used to be unemployed but now he is a meter reader. California is our beginning location.<br><br>Here is my web site; [http://inspirationpedi.com/groups/useful-guidance-for-battling-your-yeast-infection/ inspirationpedi.com] |
| The '''degree''' of a [[polynomial]] is the highest degree of its [[term (mathematics)|term]]s, when the polynomial is expressed in [[canonical form]] (i.e. as a linear combination of [[monomial]]s). The [[Degree of a monomial|degree of a term]] is the sum of the exponents of the [[Variable (mathematics)|variables]] that appear in it. The term '''[[Order of a polynomial|order]]''' refers to a different, but related concept.
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| For example, the polynomial <math>7x^2y^3 + 4x - 9</math> has three terms. (Notice, this polynomial can also be expressed as <math>7x^2y^3 + 4x^1y^0 - 9x^0y^0</math>.) The first term has a degree of 5 (the sum of the [[exponentiation|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.
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| To determine the degree of a polynomial that is not in standard form (for example <math>(x+1)^2-(x-1)^2</math>), one has to put it first in standard form by expanding the products (by [[distributivity]]) and combining the like terms; for example <math> (x+1)^2-(x-1)^2= 4x</math>, and its degree is 1, although 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. | |
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| ==Names of polynomials by degree==
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| The following names are assigned to polynomials according to their degree:<ref>{{cite web| url=http://mathforum.org/library/drmath/view/56413.html | title=Names of Polynomials | accessdate=16 September 2010 | accessdate=5 February 2012}}</ref>
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| *Degree 0 – [[constant function|constant]]
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| *Degree 1 – [[linear function|linear]]
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| *Degree 2 – [[quadratic polynomial|quadratic]]
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| *Degree 3 – [[cubic function|cubic]]
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| *Degree 4 – [[quartic function|quartic]] (or, less commonly, [[biquadratic function|biquadratic]])
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| *Degree 5 – [[quintic equation|quintic]]
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| *Degree 6 – [[sextic equation|sextic]] (or, less commonly, [[hexic equation|hexic]])
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| *Degree 7 – [[septic equation|septic]] (or, less commonly, [[heptic equation|heptic]])
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| *Degree 8 – octic
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| *Degree 9 – nonic
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| *Degree 10 – decic
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| The degree of the [[zero polynomial]] is either left undefined, or is defined to be negative (usually −1 or −∞).
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| ==Other examples==
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| *The polynomial <math>3 - 5 x + 2 x^5 - 7 x^9</math> is a nonic polynomial
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| *The polynomial <math>(y - 3)(2y + 6)(-4y - 21)</math> is a cubic polynomial
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| *The polynomial <math>(3 z^8 + z^5 - 4 z^2 + 6) + (-3 z^8 + 8 z^4 + 2 z^3 + 14 z)</math> is a quintic polynomial (as the <math>z^8</math> are cancelled out)
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| The canonical forms of the three examples above are:
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| *for <math>3 - 5 x + 2 x^5 - 7 x^9</math>, after reordering, <math>- 7 x^9 + 2 x^5 - 5 x + 3</math>;
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| *for <math>(y - 3)(2y + 6)(-4y - 21)</math>, after multiplying out and collecting terms of the same degree, <math>- 8 y^3 - 42 y^2 + 72 y + 378</math>;
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| *for <math>(3 z^8 + z^5 - 4 z^2 + 6) + (-3 z^8 + 8 z^4 + 2 z^3 + 14 z)</math>, in which the two terms of degree 8 cancel, <math>z^5 + 8 z^4 + 2 z^3 - 4 z^2 + 14 z + 6</math>.
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| ==Behavior under addition, subtraction, multiplication and function composition==
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| The degree of the sum (or difference) of two polynomials is equal to or less than the greater of their degrees, i.e.
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| :<math>\deg(P + Q) \leq \max(\deg(P),\deg(Q))</math>.
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| :<math>\deg(P - Q) \leq \max(\deg(P),\deg(Q))</math>.
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| E.g.
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| *The degree of <math>(x^3+x)+(x^2+1)=x^3+x^2+x+1</math> is 3. Note that 3 ≤ max(3, 2)
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| *The degree of <math>(x^3+x)-(x^3+x^2)=-x^2+x</math> is 2. Note that 2 ≤ max(3, 3)
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| The degree of the product of a polynomial by a non-zero [[scalar (mathematics)|scalar]] is equal to the degree of the polynomial, i.e.
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| :<math>\deg(cP)=\deg(P)</math>.
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| E.g.
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| *The degree of <math>2(x^2+3x-2)=2x^2+6x-4</math> is 2, just as the degree of <math>x^2+3x-2</math>.
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| Note that for polynomials over a [[ring (mathematics)|ring]] containing [[divisors of zero]], this is not necessarily true. For example, in [[Integers_modulo_n#Ring_of_congruence_classes|<math>\mathbf{Z}/4\mathbf{Z}</math>]], <math>\deg(1+2x) = 1</math>, but <math>\deg(2(1+2x)) = \deg(2+4x) =\deg(2) = 0</math>.
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| The [[Polynomial vector spaces|collection 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 collection is not a ring, as it is not closed under multiplication, as is seen below.)
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| The degree of the product of two polynomials over a [[field (mathematics)|field]] is the sum of their degrees
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| :<math>\deg(PQ) = \deg(P) + \deg(Q)</math>.
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| E.g.
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| *The degree of <math>(x^3+x)(x^2+1)=x^5+2x^3+x</math> is 3 + 2 = 5.
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| Note that for polynomials over an arbitrary ring, this is not necessarily true. For example, in <math>\mathbf{Z}/4\mathbf{Z}</math>, <math>\deg(2x) + \deg(1+2x) = 1 + 1 = 2</math>, but <math>\deg(2x(1+2x)) = \deg(2x) = 1</math>.
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| The degree of the composition of two polynomials over a field or [[integral domain]] is the product of their degrees
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| :<math>\deg(P \circ Q) = \deg(P)\deg(Q)</math>.
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| E.g.
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| *If <math>P = (x^3+x)</math>, <math>Q = (x^2+1)</math>, then <math>P \circ Q = P \circ (x^2+1) = (x^2+1)^3+(x^2+1) = x^6+3x^4+4x^2+2</math>, which has degree 6.
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| Note that for polynomials over an arbitrary ring, this is not necessarily true. For example, in <math>\mathbf{Z}/4\mathbf{Z}</math>, <math>\deg(2x) \deg(1+2x) = 1\cdot 1 = 1</math>, but <math>\deg(2x\circ(1+2x)) = \deg(2+4x)=\deg(2) = 0</math>.
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| ==The degree of the zero polynomial==
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| {{one source|section|date=November 2013}}
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| 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, the degree of any such function is undefined. 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.<ref>Barile, Margherita. "Zero Polynomial." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. http://mathworld.wolfram.com/ZeroPolynomial.html</ref>
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| It is convenient, however, to define the degree of the zero polynomial to be ''minus infinity'', −∞, and introduce the rules
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| :<math> \max(a,-\infty) = a, \, </math>
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| and
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| :<math> a + -\infty = -\infty. \, </math>
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| For example:
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| *The degree of the sum <math>\ (x^3+x)+(0)=x^3+x</math> is 3. Note that <math>3 \le \max(3, -\infty)</math>.
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| *The degree of the difference <math>\ x-x = 0</math> is <math>-\infty</math>. Note that <math>\ -\infty \le \max(1,1)</math>.
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| *The degree of the product <math>\ (0)(x^2+1)=0</math> is <math>\ (-\infty)+2 = -\infty</math>.
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| The price to be paid for saving the rules for computing the degree of sums and products of polynomials is that the general rule
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| :<math>\ a+b=a \quad \Rightarrow \quad b=0, \, </math>
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| breaks down when <math>\ a = -\infty</math>.
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| ==The degree computed from the function values==
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| The degree of a polynomial ''f'' can be computed by the formula
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| :<math>\deg f = \lim_{x\rarr\infty}\frac{\log |f(x)|}{\log x}.</math>
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| This formula generalizes the concept of degree to some functions that are not polynomials.
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| For example:
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| *The degree of the [[multiplicative inverse]], <math>\ 1/x</math>, is −1.
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| *The degree of the [[square root]], <math>\sqrt x </math>, is 1/2.
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| *The degree of the [[logarithm]], <math>\ \log x</math>, is 0.
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| *The degree of the [[exponential function]], <math>\ \exp x</math>, is ∞.
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| Another formula to compute the degree of ''f'' from its values is
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| :<math>\deg f = \lim_{x\to\infty}\frac{x f'(x)}{f(x)}.</math>
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| ==Extension to polynomials with two or more variables==
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| 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 ''x''<sup>2</sup>''y''<sup>2</sup> + 3''x''<sup>3</sup> + 4''y'' has degree 4, the same degree as the term ''x''<sup>2</sup>''y''<sup>2</sup>.
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| 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''.
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| :''x''<sup>2</sup>''y''<sup>2</sup> + 3''x''<sup>3</sup> + 4''y'' = (3)''x''<sup>3</sup> + (''y''<sup>2</sup>)''x''<sup>2</sup> + (4''y'') = (''x''<sup>2</sup>)''y''<sup>2</sup> + (4)''y'' + (3''x''<sup>3</sup>)
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| This polynomial has degree 3 in ''x'' and degree 2 in ''y''.
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| ==Degree function in abstract algebra==
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| Given a [[ring (mathematics)|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 (mathematics)|field]], then the polynomial ring R[''x''] is a [[principal ideal domain]] and, more importantly to our discussion here, a [[Euclidean domain]].
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| 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:
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| : deg(''f''(''x'')''g''(''x'')) = deg(''f''(''x'')) + deg(''g''(''x''))
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| For an example of why the degree function may fail over a ring that is not a field, take the following example. Let R = <math>\mathbb{Z}/4\mathbb{Z}</math>, the ring of integers [[Modular arithmetic|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'') = 2''x'' + 1. Then, ''f''(''x'')''g''(''x'') = 4''x''<sup>2</sup> + 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).
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| 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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| ==See also==
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| * [[Order of a polynomial]]
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| * [[Degree (disambiguation)#In mathematics|Degree]] — for other meanings of ''degree'' in mathematics
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| ==External links==
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| * [http://mathworld.wolfram.com/PolynomialOrder.html Polynomial Order]; Wolfram MathWorld
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| ==References==
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| <references/>
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| {{DEFAULTSORT:Degree of a Polynomial}}
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| [[Category:Polynomials]]
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