4622:; see section "Degree of a polynomial", pp. 225–226: "The product of the zero polynomial any other polynomial is always the zero polynomial, so such a property of degrees (the degree of the product is the sum of the degrees of the two factors) would not hold if one of the two polynomials were the polynomial 0. That is why we do not assign a degree to the zero polynomial."
2526:. It has no nonzero terms, and so, strictly speaking, it has no degree either. As such, its degree is usually 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.
367:
is of degree 1, even though each summand has degree 2. However, this is not needed when the polynomial is written as a product of polynomials in standard form, because the degree of a product is the sum of the degrees of the factors.
3872:
2321:
3127:
3468:
876:
4446:
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
1151:
1068:
4090:
217:
1353:
1757:
1248:
966:
2073:
2362:
4132:
3342:
1799:
1951:
1507:
1657:
736:
2899:
2785:
2660:
2482:
365:
2741:
1871:
3017:
2607:
672:
288:
122:
1421:
2956:
523:
4552:
3208:
593:
4372:
1549:
2832:
2557:
2153:
2114:
223:
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.
4652:
4575:
4526:
4495:
4468:
3519:
3300:
3244:
2979:
2855:
2517:
3606:
3277:
4407:
3546:
3177:
3373:
2414:
2385:
3683:
3577:
1999:
1979:
2158:
3042:
4805:
60:
4845:
4428:
2326:
Note that for polynomials over an arbitrary ring, the degree of the composition may be less than the product of the degrees. For example, in
1766:, the above rules may not be valid, because of cancellation that can occur when multiplying two nonzero constants. For example, in the ring
4782:
744:
4732:
3388:
5025:
976:
The degree of the sum, the product or the composition of two polynomials is strongly related to the degree of the input polynomials.
4815:
4792:
4765:
4742:
4722:
4702:
4682:
4616:
4281:
2004:
4442:
Shafarevich (2003) says of the zero polynomial: "In this case, we consider that the degree of the polynomial is undefined." (p. 27)
4317:
533:
due to degree two. There are also names for the number of terms, which are also based on Latin distributive numbers, ending in
4231:
1073:
990:
4712:
4692:
3988:
127:
1256:
984:
The degree of the sum (or difference) of two polynomials is less than or equal to the greater of their degrees; that is,
5020:
5004:
1666:
5040:
4672:
4532:
Grillet (2007) says: "The degree of the zero polynomial 0 is sometimes left undefined or is variously defined as −1 ∈
1160:
881:
4095:
For an example of why the degree function may fail over a ring that is not a field, take the following example. Let
3628:) of the polynomial is again the maximum of the degrees of all terms in the polynomial. For example, the polynomial
3551:
A more fine grained (than a simple numeric degree) description of the asymptotics of a function can be had by using
4885:
4838:
2329:
1571:
4226:
4102:
3475:
3308:
1769:
5030:
1876:
1430:
1591:
677:
4969:
2860:
2746:
2618:
2419:
297:
2676:
1808:
4999:
4994:
4989:
4251:
3928:
3556:
3145:
40:
4599:
Caldwell, William (2009), "Applying
Concept Mapping to Algebra I", in Afamasaga-Fuata'i, Karoline (ed.),
2984:
2565:
1253:
The equality always holds when the degrees of the polynomials are different. For example, the degree of
606:
229:
4979:
4959:
3140:
This formula generalizes the concept of degree to some functions that are not polynomials. For example:
545:
48:
69:
4295:
Mac Lane and
Birkhoff (1999) define "linear", "quadratic", "cubic", "quartic", and "quintic". (p. 107)
3938:
It can be shown that the degree of a polynomial over a field satisfies all of the requirements of the
1373:
5076:
5045:
4964:
4831:
3305:
The formula also gives sensible results for many combinations of such functions, e.g., the degree of
3250:
1364:
420:
404:
36:
2907:
479:
51:, the degree of the polynomial is simply the highest exponent occurring in the polynomial. The term
4858:
4209:
function is not defined for the zero element of the ring, we consider the degree of the polynomial
3920:
3033:
1578:
469:
4535:
4902:
4897:
4135:
3894:
3189:
1763:
674:
is a cubic polynomial: after multiplying out and collecting terms of the same degree, it becomes
558:
5035:
4335:
4304:
King (2009) defines "quadratic", "cubic", "quartic", "quintic", "sextic", "septic", and "octic".
1512:
2793:
2536:
2119:
2080:
4880:
4811:
4788:
4774:
4761:
4738:
4718:
4698:
4678:
4634:
4612:
4557:
4508:
4477:
4450:
4424:
4277:
4271:
3488:
3282:
3220:
2961:
2837:
2522:
Like any constant value, the value 0 can be considered as a (constant) polynomial, called the
2499:
1802:
1555:
878:
is a quintic polynomial: upon combining like terms, the two terms of degree 8 cancel, leaving
392:
3582:
3134:
5055:
4933:
4926:
4921:
4801:
4778:
4604:
3932:
3256:
427:
416:
4385:
3524:
3151:
4943:
4938:
4890:
4875:
4139:
3901:
3867:{\displaystyle x^{2}y^{2}+3x^{3}+4y=(3)x^{3}+(y^{2})x^{2}+(4y)=(x^{2})y^{2}+(4)y+(3x^{3})}
3347:
2523:
2493:
2390:
1582:
457:
440:
433:
398:
381:
4217:) = 0 to also be undefined so that it follows the rules of a norm in a Euclidean domain.
4589:
is a polynomial.) However, he excludes zero polynomials in his
Proposition 5.3. (p. 121)
2367:
2316:{\displaystyle P\circ Q=P\circ (x^{2}-1)=(x^{2}-1)^{3}+(x^{2}-1)=x^{6}-3x^{4}+4x^{2}-2,}
4914:
4909:
4823:
3562:
3552:
1984:
1964:
410:
291:
220:
4631:
Axler (1997) gives these rules and says: "The 0 polynomial is declared to have degree
5070:
4751:
4668:
464:. This should be distinguished from the names used for the number of variables, the
5050:
4313:
3028:
A number of formulae exist which will evaluate the degree of a polynomial function
1567:
4755:
4608:
3478:
to the first formula. Intuitively though, it is more about exhibiting the degree
3624:
of the exponents of the variables in the term; the degree (sometimes called the
3183:
3122:{\displaystyle \deg f=\lim _{x\rightarrow \infty }{\frac {\log |f(x)|}{\log x}}}
20:
4854:
4316:
proposed the names "sexic", "septic", "octic", "nonic", and "decic" in 1851. (
3559:, it is for example often relevant to distinguish between the growth rates of
3483:
226:
To determine the degree of a polynomial that is not in standard form, such as
28:
2529:
It is convenient, however, to define the degree of the zero polynomial to be
4984:
3214:
551:
376:
The following names are assigned to polynomials according to their degree:
4654:
so that exceptions are not needed for various reasonable results." (p. 64)
4974:
3133:
this is the exact counterpart of the method of estimating the slope in a
539:
32:
44:
3620:
For polynomials in two or more variables, the degree of a term is the
2496:
is either left undefined, or is defined to be negative (usually −1 or
1577:
More generally, the degree of the product of two polynomials over a
871:{\displaystyle (3z^{8}+z^{5}-4z^{2}+6)+(-3z^{8}+8z^{4}+2z^{3}+14z)}
465:
219:
has three terms. The first term has a degree of 5 (the sum of the
3942:
function in the euclidean domain. That is, given two polynomials
3463:{\displaystyle \deg f=\lim _{x\to \infty }{\frac {xf'(x)}{f(x)}}}
2001:
over a field or integral domain is the product of their degrees:
476:. For example, a degree two polynomial in two variables, such as
290:, one can put it in standard form by expanding the products (by
4827:
1953:, which is not equal to the sum of the degrees of the factors.
1961:
The degree of the composition of two non-constant polynomials
1562:) whose degrees are smaller than or equal to a given number
2743:
is 3. This satisfies the expected behavior, which is that
2665:
These examples illustrate how this extension satisfies the
2666:
4332:
Shafarevich (2003) says of a polynomial of degree zero,
1363:
The degree of the product of a polynomial by a non-zero
59:
but, nowadays, may refer to several other concepts (see
4737:(2nd ed.), Springer Science & Business Media,
4717:(3rd ed.), Springer Science & Business Media,
4697:(2nd ed.), Springer Science & Business Media,
4677:(2nd ed.), Springer Science & Business Media,
2981:. This satisfies the expected behavior, which is that
2857:. This satisfies the expected behavior, which is that
4601:
Concept
Mapping in Mathematics: Research into Practice
4637:
4560:
4538:
4511:
4480:
4453:
4388:
4338:
4105:
3991:
3686:
3585:
3565:
3527:
3491:
3391:
3350:
3311:
3285:
3259:
3223:
3192:
3154:
3045:
2987:
2964:
2910:
2863:
2840:
2796:
2749:
2679:
2621:
2568:
2539:
2502:
2422:
2393:
2370:
2332:
2161:
2122:
2083:
2007:
1987:
1967:
1879:
1811:
1772:
1669:
1594:
1558:
of polynomials (with coefficients from a given field
1515:
1433:
1376:
1259:
1163:
1146:{\displaystyle \deg(P-Q)\leq \max\{\deg(P),\deg(Q)\}}
1076:
1063:{\displaystyle \deg(P+Q)\leq \max\{\deg(P),\deg(Q)\}}
993:
884:
747:
680:
609:
561:
482:
300:
232:
130:
72:
4085:{\displaystyle \deg(f(x)g(x))=\deg(f(x))+\deg(g(x))}
212:{\displaystyle 7x^{2}y^{3}+4x^{1}y^{0}-9x^{0}y^{0},}
5013:
4952:
4865:
3616:
Extension to polynomials with two or more variables
1367:is equal to the degree of the polynomial; that is,
1348:{\displaystyle (x^{3}+x)+(x^{2}+1)=x^{3}+x^{2}+x+1}
35:(individual terms) with non-zero coefficients. The
4646:
4569:
4546:
4520:
4489:
4462:
4401:
4366:
4126:
4084:
3866:
3600:
3571:
3540:
3513:
3462:
3367:
3336:
3294:
3271:
3238:
3202:
3171:
3121:
3011:
2973:
2950:
2893:
2849:
2826:
2779:
2735:
2654:
2601:
2551:
2511:
2476:
2408:
2379:
2356:
2315:
2147:
2108:
2067:
1993:
1973:
1945:
1865:
1793:
1751:
1651:
1543:
1501:
1415:
1347:
1242:
1145:
1062:
960:
870:
730:
666:
587:
517:
385:
359:
282:
211:
116:
31:is the highest of the degrees of the polynomial's
4254:, with equal degree in both variables separately.
3982:individually. In fact, something stronger holds:
1752:{\displaystyle (x^{3}+x)(x^{2}+1)=x^{5}+2x^{3}+x}
4142:) because 2 × 2 = 4 ≡ 0 (mod 4). Therefore, let
4138:4. This ring is not a field (and is not even an
3931:and, more importantly to our discussion here, a
3405:
3059:
2873:
2756:
2569:
1243:{\displaystyle (x^{3}+x)-(x^{3}+x^{2})=-x^{2}+x}
1101:
1018:
961:{\displaystyle z^{5}+8z^{4}+2z^{3}-4z^{2}+14z+6}
456:Names for degree above three are based on Latin
4787:(3rd ed.), American Mathematical Society,
4194:) = 0 which is not greater than the degrees of
4876:Zero polynomial (degree undefined or −1 or −∞)
2416:(both of degree 1) is the constant polynomial
2068:{\displaystyle \deg(P\circ Q)=\deg(P)\deg(Q).}
43:that appear in it, and thus is a non-negative
4839:
4378:because if we substitute different values of
294:) and combining the like terms; for example,
8:
1140:
1104:
1057:
1021:
4276:, W. W. Norton & Company, p. 128,
3673:with coefficients which are polynomials in
3665:with coefficients which are polynomials in
2357:{\displaystyle \mathbf {Z} /4\mathbf {Z} ,}
4846:
4832:
4824:
4127:{\displaystyle \mathbb {Z} /4\mathbb {Z} }
3974:) must be larger than both the degrees of
3643:has degree 4, the same degree as the term
3608:, which would both come out as having the
3474:this second formula follows from applying
3337:{\displaystyle {\frac {1+{\sqrt {x}}}{x}}}
1794:{\displaystyle \mathbf {Z} /4\mathbf {Z} }
4810:, Springer Science & Business Media,
4760:, Springer Science & Business Media,
4714:A Concrete Introduction to Higher Algebra
4694:A Concrete Introduction to Higher Algebra
4636:
4559:
4540:
4539:
4537:
4510:
4479:
4452:
4393:
4387:
4358:
4337:
4120:
4119:
4111:
4107:
4106:
4104:
3990:
3855:
3821:
3808:
3777:
3764:
3748:
3717:
3701:
3691:
3685:
3584:
3564:
3532:
3526:
3499:
3490:
3420:
3408:
3390:
3378:Another formula to compute the degree of
3357:
3349:
3321:
3312:
3310:
3284:
3258:
3222:
3193:
3191:
3161:
3153:
3100:
3083:
3074:
3062:
3044:
2986:
2963:
2927:
2909:
2862:
2839:
2795:
2748:
2721:
2687:
2678:
2620:
2567:
2538:
2501:
2421:
2392:
2369:
2346:
2338:
2333:
2331:
2298:
2282:
2266:
2244:
2228:
2212:
2187:
2160:
2133:
2121:
2094:
2082:
2006:
1986:
1966:
1946:{\displaystyle \deg(2x(1+2x))=\deg(2x)=1}
1878:
1810:
1786:
1778:
1773:
1771:
1737:
1721:
1699:
1677:
1668:
1593:
1520:
1514:
1502:{\displaystyle 2(x^{2}+3x-2)=2x^{2}+6x-4}
1478:
1444:
1432:
1375:
1327:
1314:
1292:
1267:
1258:
1228:
1209:
1196:
1171:
1162:
1075:
992:
937:
921:
905:
889:
883:
850:
834:
818:
787:
771:
758:
746:
707:
691:
679:
608:
579:
566:
560:
509:
487:
481:
342:
317:
299:
274:
249:
231:
200:
190:
174:
164:
148:
138:
129:
90:
80:
71:
3612:degree according to the above formulae.
2155:has degree 2, then their composition is
1652:{\displaystyle \deg(PQ)=\deg(P)+\deg(Q)}
1355:is 3, and 3 = max{3, 2}.
1250:is 2, and 2 ≤ max{3, 3}.
4423:(3rd ed.), Springer, p. 100,
4382:in it, we always obtain the same value
4262:
4243:
731:{\displaystyle -8y^{3}-42y^{2}+72y+378}
2894:{\displaystyle -\infty \leq \max(1,1)}
2780:{\displaystyle 3\leq \max(3,-\infty )}
2655:{\displaystyle a+(-\infty )=-\infty .}
2559:and to introduce the arithmetic rules
2477:{\displaystyle 2x\circ (1+2x)=2+4x=2,}
1509:is 2, which is equal to the degree of
360:{\displaystyle (x+1)^{2}-(x-1)^{2}=4x}
61:Order of a polynomial (disambiguation)
4273:Mathematics From the Birth of Numbers
2736:{\displaystyle (x^{3}+x)+(0)=x^{3}+x}
1866:{\displaystyle \deg(2x)=\deg(1+2x)=1}
7:
3482:as the extra constant factor in the
972:Behavior under polynomial operations
419:(or, if all terms have even degree,
386:§ Degree of the zero polynomial
3889:Degree function in abstract algebra
3653:However, a polynomial in variables
3012:{\displaystyle -\infty =-\infty +2}
2602:{\displaystyle \max(a,-\infty )=a,}
2364:the composition of the polynomials
667:{\displaystyle (y-3)(2y+6)(-4y-21)}
283:{\displaystyle (x+1)^{2}-(x-1)^{2}}
39:is the sum of the exponents of the
4641:
4564:
4515:
4484:
4457:
3415:
3286:
3069:
3000:
2991:
2968:
2867:
2844:
2771:
2646:
2634:
2584:
2543:
2506:
1762:For polynomials over an arbitrary
1759:is 5 = 3 + 2.
595:is a "binary quadratic binomial".
525:, is called a "binary quadratic":
14:
4374:: "Such a polynomial is called a
3907:is the set of all polynomials in
3024:Computed from the function values
117:{\displaystyle 7x^{2}y^{3}+4x-9,}
4731:Grillet, Pierre Antoine (2007),
2347:
2334:
1787:
1774:
1416:{\displaystyle \deg(cP)=\deg(P)}
4444:Childs (1995) uses −1. (p. 233)
4603:, Springer, pp. 217–234,
4348:
4342:
4232:Fundamental theorem of algebra
4079:
4076:
4070:
4064:
4052:
4049:
4043:
4037:
4025:
4022:
4016:
4010:
4004:
3998:
3861:
3845:
3836:
3830:
3814:
3801:
3795:
3786:
3770:
3757:
3741:
3735:
3454:
3448:
3440:
3434:
3412:
3101:
3097:
3091:
3084:
3066:
2951:{\displaystyle (0)(x^{2}+1)=0}
2939:
2920:
2917:
2911:
2888:
2876:
2815:
2809:
2803:
2797:
2774:
2759:
2711:
2705:
2699:
2680:
2637:
2628:
2587:
2572:
2447:
2432:
2256:
2237:
2225:
2205:
2199:
2180:
2059:
2053:
2044:
2038:
2026:
2014:
1934:
1925:
1913:
1910:
1895:
1886:
1854:
1839:
1827:
1818:
1711:
1692:
1689:
1670:
1646:
1640:
1628:
1622:
1610:
1601:
1465:
1437:
1410:
1404:
1392:
1383:
1304:
1285:
1279:
1260:
1215:
1189:
1183:
1164:
1137:
1131:
1119:
1113:
1095:
1083:
1054:
1048:
1036:
1030:
1012:
1000:
865:
805:
799:
748:
661:
643:
640:
625:
622:
610:
518:{\displaystyle x^{2}+xy+y^{2}}
372:Names of polynomials by degree
339:
326:
314:
301:
271:
258:
246:
233:
55:has been used as a synonym of
1:
5036:Horner's method of evaluation
4530:Axler (1997) uses −∞. (p. 64)
3958:), the degree of the product
2790:The degree of the difference
2488:Degree of the zero polynomial
1585:is the sum of their degrees:
124:which can also be written as
4609:10.1007/978-0-387-89194-1_11
4577:, as long as deg 0 < deg
4547:{\displaystyle \mathbb {Z} }
66:For example, the polynomial
5041:Polynomial identity testing
4757:Beyond the Quartic Equation
4711:Childs, Lindsay N. (2009),
4691:Childs, Lindsay N. (1995),
4202:(which each had degree 1).
3915:. In the special case that
3669:, and also a polynomial in
3203:{\displaystyle {\sqrt {x}}}
1663:For example, the degree of
1427:For example, the degree of
1157:For example, the degree of
968:, with highest exponent 5.
738:, with highest exponent 3.
588:{\displaystyle x^{2}+y^{2}}
468:, which are based on Latin
443:(or, less commonly, heptic)
5095:
4367:{\displaystyle f(x)=a_{0}}
4250:For simplicity, this is a
3911:that have coefficients in
2904:The degree of the product
1544:{\displaystyle x^{2}+3x-2}
436:(or, less commonly, hexic)
4674:Linear Algebra Done Right
2827:{\displaystyle (x)-(x)=0}
2552:{\displaystyle -\infty ,}
2148:{\displaystyle Q=x^{2}-1}
2109:{\displaystyle P=x^{3}+x}
1572:Examples of vector spaces
4647:{\displaystyle -\infty }
4570:{\displaystyle -\infty }
4521:{\displaystyle -\infty }
4490:{\displaystyle -\infty }
4463:{\displaystyle -\infty }
3514:{\displaystyle dx^{d-1}}
3295:{\displaystyle \infty .}
3239:{\displaystyle \ \log x}
2974:{\displaystyle -\infty }
2850:{\displaystyle -\infty }
2512:{\displaystyle -\infty }
5026:Greatest common divisor
4134:, the ring of integers
3601:{\displaystyle x\log x}
4898:Quadratic function (2)
4648:
4571:
4548:
4522:
4491:
4464:
4403:
4368:
4270:Gullberg, Jan (1997),
4252:homogeneous polynomial
4128:
4086:
3929:principal ideal domain
3923:, the polynomial ring
3868:
3602:
3573:
3557:analysis of algorithms
3542:
3515:
3464:
3369:
3338:
3296:
3273:
3272:{\displaystyle \exp x}
3240:
3204:
3173:
3146:multiplicative inverse
3123:
3013:
2975:
2952:
2895:
2851:
2828:
2781:
2737:
2673:The degree of the sum
2656:
2603:
2553:
2513:
2478:
2410:
2381:
2358:
2317:
2149:
2110:
2069:
1995:
1975:
1947:
1867:
1795:
1753:
1653:
1545:
1503:
1417:
1349:
1244:
1147:
1064:
962:
872:
732:
668:
589:
549:, and (less commonly)
537:; the common ones are
529:due to two variables,
519:
361:
284:
213:
118:
4881:Constant function (0)
4807:Discourses on Algebra
4649:
4572:
4549:
4523:
4492:
4465:
4404:
4402:{\displaystyle a_{0}}
4369:
4129:
4087:
3869:
3661:, is a polynomial in
3603:
3574:
3543:
3541:{\displaystyle x^{d}}
3516:
3465:
3370:
3339:
3297:
3274:
3241:
3205:
3174:
3172:{\displaystyle \ 1/x}
3124:
3014:
2976:
2953:
2896:
2852:
2829:
2782:
2738:
2657:
2604:
2554:
2514:
2479:
2411:
2382:
2359:
2318:
2150:
2111:
2070:
1996:
1976:
1948:
1868:
1796:
1754:
1654:
1546:
1504:
1418:
1350:
1245:
1148:
1065:
963:
873:
733:
669:
590:
520:
362:
285:
214:
119:
49:univariate polynomial
5014:Tools and algorithms
4934:Quintic function (5)
4922:Quartic function (4)
4859:polynomial functions
4802:Shafarevich, Igor R.
4635:
4558:
4536:
4509:
4478:
4451:
4419:Lang, Serge (2005),
4386:
4336:
4227:Abel–Ruffini theorem
4103:
3989:
3684:
3583:
3563:
3525:
3489:
3389:
3368:{\displaystyle -1/2}
3348:
3309:
3283:
3257:
3251:exponential function
3221:
3190:
3152:
3043:
2985:
2962:
2908:
2861:
2838:
2794:
2747:
2677:
2619:
2566:
2537:
2500:
2420:
2409:{\displaystyle 1+2x}
2391:
2368:
2330:
2323:which has degree 6.
2159:
2120:
2081:
2005:
1985:
1965:
1877:
1809:
1770:
1667:
1592:
1513:
1431:
1374:
1257:
1161:
1074:
991:
882:
745:
678:
607:
559:
480:
470:distributive numbers
391:Degree 0 – non-zero
298:
230:
128:
70:
16:Mathematical concept
4944:Septic equation (7)
4939:Sextic equation (6)
4886:Linear function (1)
3382:from its values is
3034:asymptotic analysis
4910:Cubic function (3)
4903:Quadratic equation
4775:Mac Lane, Saunders
4644:
4567:
4544:
4518:
4487:
4460:
4399:
4364:
4319:Mechanics Magazine
4186:+ 1 = 1. Thus deg(
4124:
4082:
3864:
3598:
3569:
3538:
3511:
3460:
3419:
3365:
3334:
3292:
3269:
3249:The degree of the
3236:
3213:The degree of the
3200:
3182:The degree of the
3169:
3144:The degree of the
3119:
3073:
3009:
2971:
2948:
2891:
2847:
2824:
2777:
2733:
2652:
2599:
2549:
2509:
2492:The degree of the
2474:
2406:
2380:{\displaystyle 2x}
2377:
2354:
2313:
2145:
2106:
2065:
1991:
1971:
1943:
1863:
1791:
1749:
1649:
1541:
1499:
1413:
1345:
1240:
1143:
1060:
958:
868:
728:
664:
585:
515:
357:
280:
209:
114:
5064:
5063:
5005:Quasi-homogeneous
4779:Birkhoff, Garrett
4430:978-0-387-95385-4
4321:, Vol. LV, p. 171
3677:. The polynomial
3572:{\displaystyle x}
3458:
3404:
3332:
3326:
3226:
3198:
3157:
3117:
3058:
2531:negative infinity
2116:has degree 3 and
1994:{\displaystyle Q}
1974:{\displaystyle P}
1803:integers modulo 4
452:Degree 10 – decic
5084:
4927:Quartic equation
4848:
4841:
4834:
4825:
4820:
4797:
4770:
4747:
4734:Abstract Algebra
4727:
4707:
4687:
4655:
4653:
4651:
4650:
4645:
4629:
4623:
4621:
4596:
4590:
4576:
4574:
4573:
4568:
4553:
4551:
4550:
4545:
4543:
4527:
4525:
4524:
4519:
4496:
4494:
4493:
4488:
4469:
4467:
4466:
4461:
4440:
4434:
4433:
4416:
4410:
4408:
4406:
4405:
4400:
4398:
4397:
4373:
4371:
4370:
4365:
4363:
4362:
4330:
4324:
4311:
4305:
4302:
4296:
4293:
4287:
4286:
4267:
4255:
4248:
4133:
4131:
4130:
4125:
4123:
4115:
4110:
4091:
4089:
4088:
4083:
3933:Euclidean domain
3881:and degree 2 in
3877:has degree 3 in
3873:
3871:
3870:
3865:
3860:
3859:
3826:
3825:
3813:
3812:
3782:
3781:
3769:
3768:
3753:
3752:
3722:
3721:
3706:
3705:
3696:
3695:
3607:
3605:
3604:
3599:
3578:
3576:
3575:
3570:
3547:
3545:
3544:
3539:
3537:
3536:
3520:
3518:
3517:
3512:
3510:
3509:
3476:L'Hôpital's rule
3469:
3467:
3466:
3461:
3459:
3457:
3443:
3433:
3421:
3418:
3374:
3372:
3371:
3366:
3361:
3343:
3341:
3340:
3335:
3333:
3328:
3327:
3322:
3313:
3301:
3299:
3298:
3293:
3278:
3276:
3275:
3270:
3245:
3243:
3242:
3237:
3224:
3209:
3207:
3206:
3201:
3199:
3194:
3178:
3176:
3175:
3170:
3165:
3155:
3128:
3126:
3125:
3120:
3118:
3116:
3105:
3104:
3087:
3075:
3072:
3018:
3016:
3015:
3010:
2980:
2978:
2977:
2972:
2957:
2955:
2954:
2949:
2932:
2931:
2900:
2898:
2897:
2892:
2856:
2854:
2853:
2848:
2833:
2831:
2830:
2825:
2786:
2784:
2783:
2778:
2742:
2740:
2739:
2734:
2726:
2725:
2692:
2691:
2661:
2659:
2658:
2653:
2608:
2606:
2605:
2600:
2558:
2556:
2555:
2550:
2518:
2516:
2515:
2510:
2483:
2481:
2480:
2475:
2415:
2413:
2412:
2407:
2386:
2384:
2383:
2378:
2363:
2361:
2360:
2355:
2350:
2342:
2337:
2322:
2320:
2319:
2314:
2303:
2302:
2287:
2286:
2271:
2270:
2249:
2248:
2233:
2232:
2217:
2216:
2192:
2191:
2154:
2152:
2151:
2146:
2138:
2137:
2115:
2113:
2112:
2107:
2099:
2098:
2077:For example, if
2074:
2072:
2071:
2066:
2000:
1998:
1997:
1992:
1980:
1978:
1977:
1972:
1952:
1950:
1949:
1944:
1872:
1870:
1869:
1864:
1800:
1798:
1797:
1792:
1790:
1782:
1777:
1758:
1756:
1755:
1750:
1742:
1741:
1726:
1725:
1704:
1703:
1682:
1681:
1658:
1656:
1655:
1650:
1570:; for more, see
1550:
1548:
1547:
1542:
1525:
1524:
1508:
1506:
1505:
1500:
1483:
1482:
1449:
1448:
1422:
1420:
1419:
1414:
1354:
1352:
1351:
1346:
1332:
1331:
1319:
1318:
1297:
1296:
1272:
1271:
1249:
1247:
1246:
1241:
1233:
1232:
1214:
1213:
1201:
1200:
1176:
1175:
1152:
1150:
1149:
1144:
1069:
1067:
1066:
1061:
967:
965:
964:
959:
942:
941:
926:
925:
910:
909:
894:
893:
877:
875:
874:
869:
855:
854:
839:
838:
823:
822:
792:
791:
776:
775:
763:
762:
737:
735:
734:
729:
712:
711:
696:
695:
673:
671:
670:
665:
594:
592:
591:
586:
584:
583:
571:
570:
524:
522:
521:
516:
514:
513:
492:
491:
449:Degree 9 – nonic
446:Degree 8 – octic
366:
364:
363:
358:
347:
346:
322:
321:
289:
287:
286:
281:
279:
278:
254:
253:
218:
216:
215:
210:
205:
204:
195:
194:
179:
178:
169:
168:
153:
152:
143:
142:
123:
121:
120:
115:
95:
94:
85:
84:
37:degree of a term
5094:
5093:
5087:
5086:
5085:
5083:
5082:
5081:
5067:
5066:
5065:
5060:
5009:
4948:
4891:Linear equation
4861:
4852:
4818:
4800:
4795:
4773:
4768:
4750:
4745:
4730:
4725:
4710:
4705:
4690:
4685:
4667:
4664:
4659:
4658:
4633:
4632:
4630:
4626:
4619:
4598:
4597:
4593:
4556:
4555:
4534:
4533:
4531:
4529:
4507:
4506:
4501:any integer or
4476:
4475:
4449:
4448:
4445:
4443:
4441:
4437:
4431:
4418:
4417:
4413:
4389:
4384:
4383:
4354:
4334:
4333:
4331:
4327:
4312:
4308:
4303:
4299:
4294:
4290:
4284:
4269:
4268:
4264:
4259:
4258:
4249:
4245:
4240:
4223:
4140:integral domain
4101:
4100:
3987:
3986:
3902:polynomial ring
3891:
3851:
3817:
3804:
3773:
3760:
3744:
3713:
3697:
3687:
3682:
3681:
3618:
3581:
3580:
3561:
3560:
3528:
3523:
3522:
3495:
3487:
3486:
3444:
3426:
3422:
3387:
3386:
3346:
3345:
3314:
3307:
3306:
3281:
3280:
3255:
3254:
3219:
3218:
3188:
3187:
3150:
3149:
3106:
3076:
3041:
3040:
3032:. One based on
3026:
2983:
2982:
2960:
2959:
2923:
2906:
2905:
2859:
2858:
2836:
2835:
2792:
2791:
2745:
2744:
2717:
2683:
2675:
2674:
2617:
2616:
2564:
2563:
2535:
2534:
2524:zero polynomial
2498:
2497:
2494:zero polynomial
2490:
2418:
2417:
2389:
2388:
2366:
2365:
2328:
2327:
2294:
2278:
2262:
2240:
2224:
2208:
2183:
2157:
2156:
2129:
2118:
2117:
2090:
2079:
2078:
2003:
2002:
1983:
1982:
1963:
1962:
1959:
1875:
1874:
1807:
1806:
1805:, one has that
1768:
1767:
1733:
1717:
1695:
1673:
1665:
1664:
1590:
1589:
1583:integral domain
1516:
1511:
1510:
1474:
1440:
1429:
1428:
1372:
1371:
1361:
1323:
1310:
1288:
1263:
1255:
1254:
1224:
1205:
1192:
1167:
1159:
1158:
1072:
1071:
989:
988:
982:
974:
933:
917:
901:
885:
880:
879:
846:
830:
814:
783:
767:
754:
743:
742:
741:The polynomial
703:
687:
676:
675:
605:
604:
603:The polynomial
601:
575:
562:
557:
556:
505:
483:
478:
477:
458:ordinal numbers
380:Special case –
374:
338:
313:
296:
295:
270:
245:
228:
227:
196:
186:
170:
160:
144:
134:
126:
125:
86:
76:
68:
67:
17:
12:
11:
5:
5092:
5091:
5088:
5080:
5079:
5069:
5068:
5062:
5061:
5059:
5058:
5053:
5048:
5043:
5038:
5033:
5028:
5023:
5017:
5015:
5011:
5010:
5008:
5007:
5002:
4997:
4992:
4987:
4982:
4977:
4972:
4967:
4962:
4956:
4954:
4950:
4949:
4947:
4946:
4941:
4936:
4931:
4930:
4929:
4919:
4918:
4917:
4915:Cubic equation
4907:
4906:
4905:
4895:
4894:
4893:
4883:
4878:
4872:
4870:
4863:
4862:
4853:
4851:
4850:
4843:
4836:
4828:
4822:
4821:
4816:
4798:
4793:
4771:
4766:
4752:King, R. Bruce
4748:
4743:
4728:
4723:
4708:
4703:
4688:
4683:
4669:Axler, Sheldon
4663:
4660:
4657:
4656:
4643:
4640:
4624:
4617:
4591:
4566:
4563:
4542:
4517:
4514:
4486:
4483:
4459:
4456:
4435:
4429:
4411:
4396:
4392:
4361:
4357:
4353:
4350:
4347:
4344:
4341:
4325:
4306:
4297:
4288:
4282:
4261:
4260:
4257:
4256:
4242:
4241:
4239:
4236:
4235:
4234:
4229:
4222:
4219:
4122:
4118:
4114:
4109:
4093:
4092:
4081:
4078:
4075:
4072:
4069:
4066:
4063:
4060:
4057:
4054:
4051:
4048:
4045:
4042:
4039:
4036:
4033:
4030:
4027:
4024:
4021:
4018:
4015:
4012:
4009:
4006:
4003:
4000:
3997:
3994:
3890:
3887:
3875:
3874:
3863:
3858:
3854:
3850:
3847:
3844:
3841:
3838:
3835:
3832:
3829:
3824:
3820:
3816:
3811:
3807:
3803:
3800:
3797:
3794:
3791:
3788:
3785:
3780:
3776:
3772:
3767:
3763:
3759:
3756:
3751:
3747:
3743:
3740:
3737:
3734:
3731:
3728:
3725:
3720:
3716:
3712:
3709:
3704:
3700:
3694:
3690:
3617:
3614:
3597:
3594:
3591:
3588:
3568:
3553:big O notation
3535:
3531:
3508:
3505:
3502:
3498:
3494:
3472:
3471:
3456:
3453:
3450:
3447:
3442:
3439:
3436:
3432:
3429:
3425:
3417:
3414:
3411:
3407:
3403:
3400:
3397:
3394:
3364:
3360:
3356:
3353:
3331:
3325:
3320:
3317:
3303:
3302:
3291:
3288:
3268:
3265:
3262:
3247:
3235:
3232:
3229:
3211:
3197:
3180:
3179:, is −1.
3168:
3164:
3160:
3131:
3130:
3115:
3112:
3109:
3103:
3099:
3096:
3093:
3090:
3086:
3082:
3079:
3071:
3068:
3065:
3061:
3057:
3054:
3051:
3048:
3025:
3022:
3021:
3020:
3008:
3005:
3002:
2999:
2996:
2993:
2990:
2970:
2967:
2947:
2944:
2941:
2938:
2935:
2930:
2926:
2922:
2919:
2916:
2913:
2902:
2890:
2887:
2884:
2881:
2878:
2875:
2872:
2869:
2866:
2846:
2843:
2823:
2820:
2817:
2814:
2811:
2808:
2805:
2802:
2799:
2788:
2776:
2773:
2770:
2767:
2764:
2761:
2758:
2755:
2752:
2732:
2729:
2724:
2720:
2716:
2713:
2710:
2707:
2704:
2701:
2698:
2695:
2690:
2686:
2682:
2667:behavior rules
2663:
2662:
2651:
2648:
2645:
2642:
2639:
2636:
2633:
2630:
2627:
2624:
2610:
2609:
2598:
2595:
2592:
2589:
2586:
2583:
2580:
2577:
2574:
2571:
2548:
2545:
2542:
2508:
2505:
2489:
2486:
2473:
2470:
2467:
2464:
2461:
2458:
2455:
2452:
2449:
2446:
2443:
2440:
2437:
2434:
2431:
2428:
2425:
2405:
2402:
2399:
2396:
2376:
2373:
2353:
2349:
2345:
2341:
2336:
2312:
2309:
2306:
2301:
2297:
2293:
2290:
2285:
2281:
2277:
2274:
2269:
2265:
2261:
2258:
2255:
2252:
2247:
2243:
2239:
2236:
2231:
2227:
2223:
2220:
2215:
2211:
2207:
2204:
2201:
2198:
2195:
2190:
2186:
2182:
2179:
2176:
2173:
2170:
2167:
2164:
2144:
2141:
2136:
2132:
2128:
2125:
2105:
2102:
2097:
2093:
2089:
2086:
2064:
2061:
2058:
2055:
2052:
2049:
2046:
2043:
2040:
2037:
2034:
2031:
2028:
2025:
2022:
2019:
2016:
2013:
2010:
1990:
1970:
1958:
1955:
1942:
1939:
1936:
1933:
1930:
1927:
1924:
1921:
1918:
1915:
1912:
1909:
1906:
1903:
1900:
1897:
1894:
1891:
1888:
1885:
1882:
1862:
1859:
1856:
1853:
1850:
1847:
1844:
1841:
1838:
1835:
1832:
1829:
1826:
1823:
1820:
1817:
1814:
1789:
1785:
1781:
1776:
1748:
1745:
1740:
1736:
1732:
1729:
1724:
1720:
1716:
1713:
1710:
1707:
1702:
1698:
1694:
1691:
1688:
1685:
1680:
1676:
1672:
1661:
1660:
1648:
1645:
1642:
1639:
1636:
1633:
1630:
1627:
1624:
1621:
1618:
1615:
1612:
1609:
1606:
1603:
1600:
1597:
1540:
1537:
1534:
1531:
1528:
1523:
1519:
1498:
1495:
1492:
1489:
1486:
1481:
1477:
1473:
1470:
1467:
1464:
1461:
1458:
1455:
1452:
1447:
1443:
1439:
1436:
1425:
1424:
1412:
1409:
1406:
1403:
1400:
1397:
1394:
1391:
1388:
1385:
1382:
1379:
1360:
1359:Multiplication
1357:
1344:
1341:
1338:
1335:
1330:
1326:
1322:
1317:
1313:
1309:
1306:
1303:
1300:
1295:
1291:
1287:
1284:
1281:
1278:
1275:
1270:
1266:
1262:
1239:
1236:
1231:
1227:
1223:
1220:
1217:
1212:
1208:
1204:
1199:
1195:
1191:
1188:
1185:
1182:
1179:
1174:
1170:
1166:
1155:
1154:
1142:
1139:
1136:
1133:
1130:
1127:
1124:
1121:
1118:
1115:
1112:
1109:
1106:
1103:
1100:
1097:
1094:
1091:
1088:
1085:
1082:
1079:
1059:
1056:
1053:
1050:
1047:
1044:
1041:
1038:
1035:
1032:
1029:
1026:
1023:
1020:
1017:
1014:
1011:
1008:
1005:
1002:
999:
996:
981:
978:
973:
970:
957:
954:
951:
948:
945:
940:
936:
932:
929:
924:
920:
916:
913:
908:
904:
900:
897:
892:
888:
867:
864:
861:
858:
853:
849:
845:
842:
837:
833:
829:
826:
821:
817:
813:
810:
807:
804:
801:
798:
795:
790:
786:
782:
779:
774:
770:
766:
761:
757:
753:
750:
727:
724:
721:
718:
715:
710:
706:
702:
699:
694:
690:
686:
683:
663:
660:
657:
654:
651:
648:
645:
642:
639:
636:
633:
630:
627:
624:
621:
618:
615:
612:
600:
597:
582:
578:
574:
569:
565:
512:
508:
504:
501:
498:
495:
490:
486:
454:
453:
450:
447:
444:
437:
430:
424:
413:
407:
401:
395:
389:
373:
370:
356:
353:
350:
345:
341:
337:
334:
331:
328:
325:
320:
316:
312:
309:
306:
303:
292:distributivity
277:
273:
269:
266:
263:
260:
257:
252:
248:
244:
241:
238:
235:
208:
203:
199:
193:
189:
185:
182:
177:
173:
167:
163:
159:
156:
151:
147:
141:
137:
133:
113:
110:
107:
104:
101:
98:
93:
89:
83:
79:
75:
15:
13:
10:
9:
6:
4:
3:
2:
5090:
5089:
5078:
5075:
5074:
5072:
5057:
5056:Gröbner basis
5054:
5052:
5049:
5047:
5044:
5042:
5039:
5037:
5034:
5032:
5029:
5027:
5024:
5022:
5021:Factorization
5019:
5018:
5016:
5012:
5006:
5003:
5001:
4998:
4996:
4993:
4991:
4988:
4986:
4983:
4981:
4978:
4976:
4973:
4971:
4968:
4966:
4963:
4961:
4958:
4957:
4955:
4953:By properties
4951:
4945:
4942:
4940:
4937:
4935:
4932:
4928:
4925:
4924:
4923:
4920:
4916:
4913:
4912:
4911:
4908:
4904:
4901:
4900:
4899:
4896:
4892:
4889:
4888:
4887:
4884:
4882:
4879:
4877:
4874:
4873:
4871:
4869:
4864:
4860:
4856:
4849:
4844:
4842:
4837:
4835:
4830:
4829:
4826:
4819:
4817:9783540422532
4813:
4809:
4808:
4803:
4799:
4796:
4794:9780821816462
4790:
4786:
4785:
4780:
4776:
4772:
4769:
4767:9780817648497
4763:
4759:
4758:
4753:
4749:
4746:
4744:9780387715681
4740:
4736:
4735:
4729:
4726:
4724:9780387745275
4720:
4716:
4715:
4709:
4706:
4704:9780387989990
4700:
4696:
4695:
4689:
4686:
4684:9780387982595
4680:
4676:
4675:
4670:
4666:
4665:
4661:
4638:
4628:
4625:
4620:
4618:9780387891941
4614:
4610:
4606:
4602:
4595:
4592:
4588:
4584:
4580:
4561:
4512:
4504:
4500:
4481:
4473:
4454:
4439:
4436:
4432:
4426:
4422:
4415:
4412:
4394:
4390:
4381:
4377:
4359:
4355:
4351:
4345:
4339:
4329:
4326:
4322:
4320:
4315:
4310:
4307:
4301:
4298:
4292:
4289:
4285:
4283:9780393040029
4279:
4275:
4274:
4266:
4263:
4253:
4247:
4244:
4237:
4233:
4230:
4228:
4225:
4224:
4220:
4218:
4216:
4212:
4208:
4203:
4201:
4197:
4193:
4189:
4185:
4181:
4177:
4173:
4169:
4165:
4161:
4157:
4153:
4149:
4145:
4141:
4137:
4116:
4112:
4098:
4073:
4067:
4061:
4058:
4055:
4046:
4040:
4034:
4031:
4028:
4019:
4013:
4007:
4001:
3995:
3992:
3985:
3984:
3983:
3981:
3977:
3973:
3969:
3965:
3961:
3957:
3953:
3949:
3945:
3941:
3936:
3934:
3930:
3926:
3922:
3918:
3914:
3910:
3906:
3903:
3899:
3896:
3888:
3886:
3884:
3880:
3856:
3852:
3848:
3842:
3839:
3833:
3827:
3822:
3818:
3809:
3805:
3798:
3792:
3789:
3783:
3778:
3774:
3765:
3761:
3754:
3749:
3745:
3738:
3732:
3729:
3726:
3723:
3718:
3714:
3710:
3707:
3702:
3698:
3692:
3688:
3680:
3679:
3678:
3676:
3672:
3668:
3664:
3660:
3656:
3651:
3649:
3646:
3642:
3638:
3634:
3631:
3627:
3623:
3615:
3613:
3611:
3595:
3592:
3589:
3586:
3566:
3558:
3554:
3549:
3533:
3529:
3506:
3503:
3500:
3496:
3492:
3485:
3481:
3477:
3451:
3445:
3437:
3430:
3427:
3423:
3409:
3401:
3398:
3395:
3392:
3385:
3384:
3383:
3381:
3376:
3362:
3358:
3354:
3351:
3329:
3323:
3318:
3315:
3289:
3266:
3263:
3260:
3252:
3248:
3233:
3230:
3227:
3216:
3212:
3195:
3185:
3181:
3166:
3162:
3158:
3147:
3143:
3142:
3141:
3138:
3136:
3113:
3110:
3107:
3094:
3088:
3080:
3077:
3063:
3055:
3052:
3049:
3046:
3039:
3038:
3037:
3035:
3031:
3023:
3006:
3003:
2997:
2994:
2988:
2965:
2945:
2942:
2936:
2933:
2928:
2924:
2914:
2903:
2885:
2882:
2879:
2870:
2864:
2841:
2821:
2818:
2812:
2806:
2800:
2789:
2768:
2765:
2762:
2753:
2750:
2730:
2727:
2722:
2718:
2714:
2708:
2702:
2696:
2693:
2688:
2684:
2672:
2671:
2670:
2668:
2649:
2643:
2640:
2631:
2625:
2622:
2615:
2614:
2613:
2596:
2593:
2590:
2581:
2578:
2575:
2562:
2561:
2560:
2546:
2540:
2532:
2527:
2525:
2520:
2503:
2495:
2487:
2485:
2484:of degree 0.
2471:
2468:
2465:
2462:
2459:
2456:
2453:
2450:
2444:
2441:
2438:
2435:
2429:
2426:
2423:
2403:
2400:
2397:
2394:
2374:
2371:
2351:
2343:
2339:
2324:
2310:
2307:
2304:
2299:
2295:
2291:
2288:
2283:
2279:
2275:
2272:
2267:
2263:
2259:
2253:
2250:
2245:
2241:
2234:
2229:
2221:
2218:
2213:
2209:
2202:
2196:
2193:
2188:
2184:
2177:
2174:
2171:
2168:
2165:
2162:
2142:
2139:
2134:
2130:
2126:
2123:
2103:
2100:
2095:
2091:
2087:
2084:
2075:
2062:
2056:
2050:
2047:
2041:
2035:
2032:
2029:
2023:
2020:
2017:
2011:
2008:
1988:
1968:
1956:
1954:
1940:
1937:
1931:
1928:
1922:
1919:
1916:
1907:
1904:
1901:
1898:
1892:
1889:
1883:
1880:
1860:
1857:
1851:
1848:
1845:
1842:
1836:
1833:
1830:
1824:
1821:
1815:
1812:
1804:
1783:
1779:
1765:
1760:
1746:
1743:
1738:
1734:
1730:
1727:
1722:
1718:
1714:
1708:
1705:
1700:
1696:
1686:
1683:
1678:
1674:
1643:
1637:
1634:
1631:
1625:
1619:
1616:
1613:
1607:
1604:
1598:
1595:
1588:
1587:
1586:
1584:
1580:
1575:
1573:
1569:
1565:
1561:
1557:
1552:
1538:
1535:
1532:
1529:
1526:
1521:
1517:
1496:
1493:
1490:
1487:
1484:
1479:
1475:
1471:
1468:
1462:
1459:
1456:
1453:
1450:
1445:
1441:
1434:
1407:
1401:
1398:
1395:
1389:
1386:
1380:
1377:
1370:
1369:
1368:
1366:
1358:
1356:
1342:
1339:
1336:
1333:
1328:
1324:
1320:
1315:
1311:
1307:
1301:
1298:
1293:
1289:
1282:
1276:
1273:
1268:
1264:
1251:
1237:
1234:
1229:
1225:
1221:
1218:
1210:
1206:
1202:
1197:
1193:
1186:
1180:
1177:
1172:
1168:
1134:
1128:
1125:
1122:
1116:
1110:
1107:
1098:
1092:
1089:
1086:
1080:
1077:
1051:
1045:
1042:
1039:
1033:
1027:
1024:
1015:
1009:
1006:
1003:
997:
994:
987:
986:
985:
979:
977:
971:
969:
955:
952:
949:
946:
943:
938:
934:
930:
927:
922:
918:
914:
911:
906:
902:
898:
895:
890:
886:
862:
859:
856:
851:
847:
843:
840:
835:
831:
827:
824:
819:
815:
811:
808:
802:
796:
793:
788:
784:
780:
777:
772:
768:
764:
759:
755:
751:
739:
725:
722:
719:
716:
713:
708:
704:
700:
697:
692:
688:
684:
681:
658:
655:
652:
649:
646:
637:
634:
631:
628:
619:
616:
613:
598:
596:
580:
576:
572:
567:
563:
554:
553:
548:
547:
542:
541:
536:
532:
528:
510:
506:
502:
499:
496:
493:
488:
484:
475:
472:, and end in
471:
467:
463:
460:, and end in
459:
451:
448:
445:
442:
438:
435:
431:
429:
425:
422:
418:
414:
412:
408:
406:
402:
400:
396:
394:
390:
387:
383:
379:
378:
377:
371:
369:
354:
351:
348:
343:
335:
332:
329:
323:
318:
310:
307:
304:
293:
275:
267:
264:
261:
255:
250:
242:
239:
236:
224:
222:
206:
201:
197:
191:
187:
183:
180:
175:
171:
165:
161:
157:
154:
149:
145:
139:
135:
131:
111:
108:
105:
102:
99:
96:
91:
87:
81:
77:
73:
64:
62:
58:
54:
50:
46:
42:
38:
34:
30:
26:
22:
5051:Discriminant
4970:Multivariate
4867:
4806:
4783:
4756:
4733:
4713:
4693:
4673:
4627:
4600:
4594:
4586:
4582:
4578:
4502:
4498:
4471:
4438:
4420:
4414:
4379:
4375:
4328:
4318:
4314:James Cockle
4309:
4300:
4291:
4272:
4265:
4246:
4214:
4210:
4206:
4204:
4199:
4195:
4191:
4187:
4183:
4179:
4175:
4171:
4167:
4163:
4159:
4155:
4151:
4147:
4143:
4096:
4094:
3979:
3975:
3971:
3967:
3963:
3959:
3955:
3951:
3947:
3943:
3939:
3937:
3924:
3916:
3912:
3908:
3904:
3897:
3892:
3882:
3878:
3876:
3674:
3670:
3666:
3662:
3658:
3654:
3652:
3647:
3644:
3640:
3636:
3632:
3629:
3626:total degree
3625:
3621:
3619:
3609:
3550:
3479:
3473:
3379:
3377:
3304:
3139:
3135:log–log plot
3132:
3029:
3027:
2664:
2611:
2530:
2528:
2521:
2491:
2325:
2076:
1960:
1761:
1662:
1576:
1568:vector space
1563:
1559:
1553:
1426:
1362:
1252:
1156:
983:
975:
740:
602:
550:
544:
538:
534:
530:
526:
473:
461:
455:
375:
225:
65:
56:
52:
24:
18:
5077:Polynomials
5000:Homogeneous
4995:Square-free
4990:Irreducible
4855:Polynomials
4162:+ 1. Then,
3184:square root
1957:Composition
439:Degree 7 –
432:Degree 6 –
426:Degree 5 –
421:biquadratic
415:Degree 4 –
409:Degree 3 –
403:Degree 2 –
397:Degree 1 –
21:mathematics
4960:Univariate
4662:References
4409:." (p. 23)
4205:Since the
3919:is also a
3484:derivative
1554:Thus, the
29:polynomial
5046:Resultant
4985:Trinomial
4965:Bivariate
4642:∞
4639:−
4565:∞
4562:−
4516:∞
4513:−
4485:∞
4482:−
4458:∞
4455:−
4062:
4035:
3996:
3593:
3555:. In the
3504:−
3416:∞
3413:→
3396:
3352:−
3287:∞
3264:
3231:
3215:logarithm
3210:, is 1/2.
3111:
3081:
3070:∞
3067:→
3050:
3001:∞
2998:−
2992:∞
2989:−
2969:∞
2966:−
2871:≤
2868:∞
2865:−
2845:∞
2842:−
2807:−
2772:∞
2769:−
2754:≤
2647:∞
2644:−
2635:∞
2632:−
2585:∞
2582:−
2544:∞
2541:−
2507:∞
2504:−
2430:∘
2305:−
2273:−
2251:−
2219:−
2194:−
2178:∘
2166:∘
2140:−
2051:
2036:
2021:∘
2012:
1923:
1884:
1837:
1816:
1638:
1620:
1599:
1536:−
1494:−
1460:−
1402:
1381:
1222:−
1187:−
1129:
1111:
1099:≤
1090:−
1081:
1046:
1028:
1016:≤
998:
928:−
809:−
778:−
698:−
682:−
656:−
647:−
617:−
552:trinomial
531:quadratic
405:quadratic
333:−
324:−
265:−
256:−
181:−
106:−
41:variables
33:monomials
5071:Category
5031:Division
4980:Binomial
4975:Monomial
4804:(2003),
4781:(1999),
4754:(2009),
4671:(1997),
4581:for all
4376:constant
4221:See also
3893:Given a
3431:′
1566:forms a
980:Addition
599:Examples
546:binomial
540:monomial
393:constant
388:, below)
47:. For a
4784:Algebra
4585:≠ 0." (
4421:Algebra
3246:, is 0.
2669:above:
555:; thus
535:-nomial
428:quintic
417:quartic
45:integer
4868:degree
4814:
4791:
4764:
4741:
4721:
4701:
4681:
4615:
4554:or as
4427:
4280:
4136:modulo
3950:) and
3900:, the
3225:
3156:
1873:, but
1581:or an
1365:scalar
527:binary
441:septic
434:sextic
399:linear
221:powers
57:degree
25:degree
23:, the
4238:Notes
4178:) = 4
4158:) = 2
3927:is a
3921:field
3279:, is
1579:field
466:arity
411:cubic
384:(see
53:order
27:of a
4857:and
4812:ISBN
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4497:for
4425:ISBN
4278:ISBN
4207:norm
4198:and
4150:) =
3978:and
3940:norm
3895:ring
3657:and
3610:same
3579:and
2612:and
2387:and
1981:and
1764:ring
1070:and
474:-ary
382:zero
4866:By
4605:doi
4182:+ 4
4059:deg
4032:deg
3993:deg
3639:+ 4
3635:+ 3
3622:sum
3590:log
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3406:lim
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3261:exp
3228:log
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2009:deg
1920:deg
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1556:set
1399:deg
1378:deg
1126:deg
1108:deg
1102:max
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1025:deg
1019:max
995:deg
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860:14
717:72
701:42
659:21
543:,
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4211:f
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4196:f
4192:g
4190:⋅
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3790:4
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3784:+
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3758:(
3755:+
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2889:)
2886:1
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2709:0
2706:(
2703:+
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2697:x
2694:+
2689:3
2685:x
2681:(
2650:.
2641:=
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2629:(
2626:+
2623:a
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2594:a
2591:=
2588:)
2579:,
2576:a
2573:(
2547:,
2472:,
2469:2
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2463:x
2460:4
2457:+
2454:2
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2448:)
2445:x
2442:2
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2436:1
2433:(
2427:x
2424:2
2404:x
2401:2
2398:+
2395:1
2375:x
2372:2
2352:,
2348:Z
2344:4
2340:/
2335:Z
2311:,
2308:2
2300:2
2296:x
2292:4
2289:+
2284:4
2280:x
2276:3
2268:6
2264:x
2260:=
2257:)
2254:1
2246:2
2242:x
2238:(
2235:+
2230:3
2226:)
2222:1
2214:2
2210:x
2206:(
2203:=
2200:)
2197:1
2189:2
2185:x
2181:(
2175:P
2172:=
2169:Q
2163:P
2143:1
2135:2
2131:x
2127:=
2124:Q
2104:x
2101:+
2096:3
2092:x
2088:=
2085:P
2063:.
2060:)
2057:Q
2054:(
2045:)
2042:P
2039:(
2030:=
2027:)
2024:Q
2018:P
2015:(
1989:Q
1969:P
1941:1
1938:=
1935:)
1932:x
1929:2
1926:(
1917:=
1914:)
1911:)
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1896:(
1893:x
1890:2
1887:(
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1822:2
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1780:/
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1709:1
1706:+
1701:2
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1693:(
1690:)
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1623:(
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1608:Q
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1602:(
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1560:F
1539:2
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1527:+
1522:2
1518:x
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1488:6
1485:+
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1454:3
1451:+
1446:2
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1435:2
1423:.
1411:)
1408:P
1405:(
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1390:P
1387:c
1384:(
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1283:+
1280:)
1277:x
1274:+
1269:3
1265:x
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1211:2
1207:x
1203:+
1198:3
1194:x
1190:(
1184:)
1181:x
1178:+
1173:3
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1165:(
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1135:Q
1132:(
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1117:P
1114:(
1105:{
1096:)
1093:Q
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1058:}
1055:)
1052:Q
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1034:P
1031:(
1022:{
1013:)
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1001:(
956:6
953:+
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915:2
912:+
907:4
903:z
899:8
896:+
891:5
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863:z
857:+
852:3
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828:8
825:+
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812:3
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797:6
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765:+
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752:3
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714:+
709:2
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693:3
689:y
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653:y
650:4
644:(
641:)
638:6
635:+
632:y
629:2
626:(
623:)
620:3
614:y
611:(
581:2
577:y
573:+
568:2
564:x
511:2
507:y
503:+
500:y
497:x
494:+
489:2
485:x
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349:=
344:2
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336:1
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327:(
319:2
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311:1
308:+
305:x
302:(
276:2
272:)
268:1
262:x
259:(
251:2
247:)
243:1
240:+
237:x
234:(
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202:0
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192:0
188:x
184:9
176:0
172:y
166:1
162:x
158:4
155:+
150:3
146:y
140:2
136:x
132:7
112:,
109:9
103:x
100:4
97:+
92:3
88:y
82:2
78:x
74:7
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