4633:; 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."
2537:. 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.
378:
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.
3883:
2332:
3138:
3479:
887:
4457:
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
1162:
1079:
4101:
228:
1364:
1768:
1259:
977:
2084:
2373:
4143:
3353:
1810:
1962:
1518:
1668:
747:
2910:
2796:
2671:
2493:
376:
2752:
1882:
3028:
2618:
683:
299:
133:
1432:
2967:
534:
4563:
3219:
604:
4383:
1560:
2843:
2568:
2164:
2125:
234:
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.
4663:
4586:
4537:
4506:
4479:
3530:
3311:
3255:
2990:
2866:
2528:
3617:
3288:
4418:
3557:
3188:
3384:
2425:
2396:
3694:
3588:
2010:
1990:
2169:
3053:
4816:
2337:
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
1777:, the above rules may not be valid, because of cancellation that can occur when multiplying two nonzero constants. For example, in the ring
4793:
755:
4743:
3399:
987:
The degree of the sum, the product or the composition of two polynomials is strongly related to the degree of the input polynomials.
2015:
4453:
Shafarevich (2003) says of the zero polynomial: "In this case, we consider that the degree of the polynomial is undefined." (p. 27)
4328:
544:
due to degree two. There are also names for the number of terms, which are also based on Latin distributive numbers, ending in
71:
1582:
4856:
4439:
1084:
1001:
4723:
4703:
3999:
138:
1267:
995:
The degree of the sum (or difference) of two polynomials is less than or equal to the greater of their degrees; that is,
5036:
1677:
4683:
4543:
Grillet (2007) says: "The degree of the zero polynomial 0 is sometimes left undefined or is variously defined as −1 ∈
1171:
892:
4826:
4803:
4776:
4753:
4733:
4713:
4693:
4627:
4292:
4106:
For an example of why the degree function may fail over a ring that is not a field, take the following example. Let
3639:) of the polynomial is again the maximum of the degrees of all terms in the polynomial. For example, the polynomial
3562:
A more fine grained (than a simple numeric degree) description of the asymptotics of a function can be had by using
2340:
4113:
3319:
1780:
4242:
1887:
1441:
1602:
688:
2871:
2757:
2629:
2430:
308:
5031:
5015:
2687:
1819:
17:
5051:
4610:
Caldwell, William (2009), "Applying
Concept Mapping to Algebra I", in Afamasaga-Fuata'i, Karoline (ed.),
2995:
2576:
1264:
The equality always holds when the degrees of the polynomials are different. For example, the degree of
617:
240:
3151:
This formula generalizes the concept of degree to some functions that are not polynomials. For example:
4896:
80:
4306:
Mac Lane and
Birkhoff (1999) define "linear", "quadratic", "cubic", "quartic", and "quintic". (p. 107)
3949:
It can be shown that the degree of a polynomial over a field satisfies all of the requirements of the
1384:
4849:
3316:
The formula also gives sensible results for many combinations of such functions, e.g., the degree of
3486:
2918:
490:
62:, the degree of the polynomial is simply the highest exponent occurring in the polynomial. The term
5041:
4237:
4220:
function is not defined for the zero element of the ring, we consider the degree of the polynomial
4546:
4980:
3200:
685:
is a cubic polynomial: after multiplying out and collecting terms of the same degree, it becomes
569:
4346:
4315:
King (2009) defines "quadratic", "cubic", "quartic", "quintic", "sextic", "septic", and "octic".
1523:
5010:
5005:
5000:
4262:
3939:
3567:
3156:
2804:
2547:
2130:
2091:
51:
4645:
4568:
4519:
4488:
4461:
4282:
3499:
3293:
3231:
2972:
2848:
2533:
Like any constant value, the value 0 can be considered as a (constant) polynomial, called the
2510:
889:
is a quintic polynomial: upon combining like terms, the two terms of degree 8 cancel, leaving
4990:
4970:
3593:
556:
59:
3267:
5087:
5056:
4975:
4842:
4396:
3535:
3261:
3162:
1375:
431:
415:
47:
3878:{\displaystyle x^{2}y^{2}+3x^{3}+4y=(3)x^{3}+(y^{2})x^{2}+(4y)=(x^{2})y^{2}+(4)y+(3x^{3})}
3358:
2401:
8:
4869:
4228:) = 0 to also be undefined so that it follows the rules of a norm in a Euclidean domain.
3931:
3044:
1589:
480:
4600:
is a polynomial.) However, he excludes zero polynomials in his
Proposition 5.3. (p. 121)
2378:
2327:{\displaystyle P\circ Q=P\circ (x^{2}-1)=(x^{2}-1)^{3}+(x^{2}-1)=x^{6}-3x^{4}+4x^{2}-2,}
5046:
4913:
4908:
4834:
4146:
3905:
3573:
1995:
1975:
1774:
4642:
Axler (1997) gives these rules and says: "The 0 polynomial is declared to have degree
4891:
4822:
4799:
4785:
4772:
4749:
4729:
4709:
4689:
4623:
4435:
4288:
1813:
1566:
403:
475:. This should be distinguished from the names used for the number of variables, the
4944:
4937:
4932:
4812:
4789:
4615:
3943:
3145:
3039:
A number of formulae exist which will evaluate the degree of a polynomial function
438:
427:
5066:
4954:
4949:
4901:
4886:
4766:
4619:
4150:
3912:
3489:
to the first formula. Intuitively though, it is more about exhibiting the degree
2534:
2504:
1593:
468:
451:
444:
409:
392:
3635:
of the exponents of the variables in the term; the degree (sometimes called the
3133:{\displaystyle \deg f=\lim _{x\rightarrow \infty }{\frac {\log |f(x)|}{\log x}}}
4925:
4920:
3563:
421:
302:
231:
4327:
proposed the names "sexic", "septic", "octic", "nonic", and "decic" in 1851. (
3570:, it is for example often relevant to distinguish between the growth rates of
237:
To determine the degree of a polynomial that is not in standard form, such as
5081:
4762:
4679:
2540:
It is convenient, however, to define the degree of the zero polynomial to be
5061:
4324:
1578:
387:
The following names are assigned to polynomials according to their degree:
4665:
so that exceptions are not needed for various reasonable results." (p. 64)
3194:
3144:
this is the exact counterpart of the method of estimating the slope in a
31:
4865:
3494:
39:
3631:
For polynomials in two or more variables, the degree of a term is the
3626:
2507:
is either left undefined, or is defined to be negative (usually −1 or
4995:
3225:
562:
1588:
More generally, the degree of the product of two polynomials over a
4985:
882:{\displaystyle (3z^{8}+z^{5}-4z^{2}+6)+(-3z^{8}+8z^{4}+2z^{3}+14z)}
550:
43:
230:
has three terms. The first term has a degree of 5 (the sum of the
3953:
function in the euclidean domain. That is, given two polynomials
3474:{\displaystyle \deg f=\lim _{x\to \infty }{\frac {xf'(x)}{f(x)}}}
2012:
over a field or integral domain is the product of their degrees:
487:. For example, a degree two polynomial in two variables, such as
55:
301:, one can put it in standard form by expanding the products (by
1964:, which is not equal to the sum of the degrees of the factors.
1972:
The degree of the composition of two non-constant polynomials
476:
1573:) whose degrees are smaller than or equal to a given number
2754:
is 3. This satisfies the expected behavior, which is that
2676:
These examples illustrate how this extension satisfies the
2677:
4864:
4343:
Shafarevich (2003) says of a polynomial of degree zero,
1374:
The degree of the product of a polynomial by a non-zero
70:
but, nowadays, may refer to several other concepts (see
4748:(2nd ed.), Springer Science & Business Media,
4728:(3rd ed.), Springer Science & Business Media,
4708:(2nd ed.), Springer Science & Business Media,
4688:(2nd ed.), Springer Science & Business Media,
2992:. This satisfies the expected behavior, which is that
2868:. This satisfies the expected behavior, which is that
982:
4612:
Concept
Mapping in Mathematics: Research into Practice
3899:
4648:
4571:
4549:
4522:
4491:
4464:
4399:
4349:
4116:
4002:
3697:
3596:
3576:
3538:
3502:
3402:
3361:
3322:
3296:
3270:
3234:
3203:
3165:
3056:
2998:
2975:
2921:
2874:
2851:
2807:
2760:
2690:
2632:
2579:
2550:
2513:
2433:
2404:
2381:
2343:
2172:
2133:
2094:
2018:
1998:
1978:
1890:
1822:
1783:
1680:
1605:
1569:
of polynomials (with coefficients from a given field
1526:
1444:
1387:
1270:
1174:
1157:{\displaystyle \deg(P-Q)\leq \max\{\deg(P),\deg(Q)\}}
1087:
1074:{\displaystyle \deg(P+Q)\leq \max\{\deg(P),\deg(Q)\}}
1004:
895:
758:
691:
620:
572:
493:
311:
243:
141:
83:
4096:{\displaystyle \deg(f(x)g(x))=\deg(f(x))+\deg(g(x))}
3034:
223:{\displaystyle 7x^{2}y^{3}+4x^{1}y^{0}-9x^{0}y^{0},}
3627:
Extension to polynomials with two or more variables
1378:is equal to the degree of the polynomial; that is,
1359:{\displaystyle (x^{3}+x)+(x^{2}+1)=x^{3}+x^{2}+x+1}
46:(individual terms) with non-zero coefficients. The
4657:
4580:
4557:
4531:
4500:
4473:
4412:
4377:
4137:
4095:
3877:
3611:
3582:
3551:
3524:
3473:
3378:
3347:
3305:
3282:
3249:
3213:
3182:
3132:
3022:
2984:
2961:
2904:
2860:
2837:
2790:
2746:
2665:
2612:
2562:
2522:
2487:
2419:
2390:
2367:
2326:
2158:
2119:
2078:
2004:
1984:
1956:
1876:
1804:
1762:
1662:
1554:
1512:
1426:
1358:
1253:
1156:
1073:
971:
881:
741:
677:
598:
528:
396:
370:
293:
222:
127:
42:is the highest of the degrees of the polynomial's
4265:, with equal degree in both variables separately.
3993:individually. In fact, something stronger holds:
1763:{\displaystyle (x^{3}+x)(x^{2}+1)=x^{5}+2x^{3}+x}
382:
5079:
4784:
4153:) because 2 × 2 = 4 ≡ 0 (mod 4). Therefore, let
4149:4. This ring is not a field (and is not even an
3942:and, more importantly to our discussion here, a
3416:
3070:
2884:
2767:
2580:
2498:
1254:{\displaystyle (x^{3}+x)-(x^{3}+x^{2})=-x^{2}+x}
1112:
1029:
972:{\displaystyle z^{5}+8z^{4}+2z^{3}-4z^{2}+14z+6}
467:Names for degree above three are based on Latin
4798:(3rd ed.), American Mathematical Society,
4205:) = 0 which is not greater than the degrees of
4887:Zero polynomial (degree undefined or −1 or −∞)
2427:(both of degree 1) is the constant polynomial
2079:{\displaystyle \deg(P\circ Q)=\deg(P)\deg(Q).}
54:that appear in it, and thus is a non-negative
4850:
4389:because if we substitute different values of
305:) and combining the like terms; for example,
1151:
1115:
1068:
1032:
4811:
4287:, W. W. Norton & Company, p. 128,
3684:with coefficients which are polynomials in
3676:with coefficients which are polynomials in
2368:{\displaystyle \mathbf {Z} /4\mathbf {Z} ,}
4857:
4843:
4138:{\displaystyle \mathbb {Z} /4\mathbb {Z} }
3985:) must be larger than both the degrees of
3654:has degree 4, the same degree as the term
3619:, which would both come out as having the
3485:this second formula follows from applying
3348:{\displaystyle {\frac {1+{\sqrt {x}}}{x}}}
1805:{\displaystyle \mathbf {Z} /4\mathbf {Z} }
4821:, Springer Science & Business Media,
4771:, Springer Science & Business Media,
4725:A Concrete Introduction to Higher Algebra
4705:A Concrete Introduction to Higher Algebra
4551:
4131:
4118:
3389:Another formula to compute the degree of
1957:{\displaystyle \deg(2x(1+2x))=\deg(2x)=1}
1513:{\displaystyle 2(x^{2}+3x-2)=2x^{2}+6x-4}
4609:
4280:
3623:degree according to the above formulae.
2166:has degree 2, then their composition is
1663:{\displaystyle \deg(PQ)=\deg(P)+\deg(Q)}
1366:is 3, and 3 = max{3, 2}.
1261:is 2, and 2 ≤ max{3, 3}.
4741:
4434:(3rd ed.), Springer, p. 100,
4393:in it, we always obtain the same value
742:{\displaystyle -8y^{3}-42y^{2}+72y+378}
14:
5080:
4721:
4701:
2905:{\displaystyle -\infty \leq \max(1,1)}
2791:{\displaystyle 3\leq \max(3,-\infty )}
2666:{\displaystyle a+(-\infty )=-\infty .}
2570:and to introduce the arithmetic rules
2488:{\displaystyle 2x\circ (1+2x)=2+4x=2,}
1520:is 2, which is equal to the degree of
371:{\displaystyle (x+1)^{2}-(x-1)^{2}=4x}
72:Order of a polynomial (disambiguation)
4838:
4678:
4284:Mathematics From the Birth of Numbers
2747:{\displaystyle (x^{3}+x)+(0)=x^{3}+x}
1877:{\displaystyle \deg(2x)=\deg(1+2x)=1}
4761:
3493:as the extra constant factor in the
983:Behavior under polynomial operations
430:(or, if all terms have even degree,
397:§ Degree of the zero polynomial
3900:Degree function in abstract algebra
3664:However, a polynomial in variables
3023:{\displaystyle -\infty =-\infty +2}
2613:{\displaystyle \max(a,-\infty )=a,}
2375:the composition of the polynomials
678:{\displaystyle (y-3)(2y+6)(-4y-21)}
294:{\displaystyle (x+1)^{2}-(x-1)^{2}}
50:is the sum of the exponents of the
24:
4652:
4575:
4526:
4495:
4468:
3426:
3297:
3080:
3011:
3002:
2979:
2878:
2855:
2782:
2657:
2645:
2595:
2554:
2517:
1773:For polynomials over an arbitrary
1770:is 5 = 3 + 2.
606:is a "binary quadratic binomial".
536:, is called a "binary quadratic":
25:
5099:
4385:: "Such a polynomial is called a
3918:is the set of all polynomials in
3035:Computed from the function values
1369:
128:{\displaystyle 7x^{2}y^{3}+4x-9,}
4742:Grillet, Pierre Antoine (2007),
2358:
2345:
1798:
1785:
1427:{\displaystyle \deg(cP)=\deg(P)}
4636:
4455:Childs (1995) uses −1. (p. 233)
4614:, Springer, pp. 217–234,
4603:
4447:
4423:
4359:
4353:
4337:
4318:
4309:
4300:
4274:
4255:
4243:Fundamental theorem of algebra
4090:
4087:
4081:
4075:
4063:
4060:
4054:
4048:
4036:
4033:
4027:
4021:
4015:
4009:
3872:
3856:
3847:
3841:
3825:
3812:
3806:
3797:
3781:
3768:
3752:
3746:
3465:
3459:
3451:
3445:
3423:
3112:
3108:
3102:
3095:
3077:
2962:{\displaystyle (0)(x^{2}+1)=0}
2950:
2931:
2928:
2922:
2899:
2887:
2826:
2820:
2814:
2808:
2785:
2770:
2722:
2716:
2710:
2691:
2648:
2639:
2598:
2583:
2458:
2443:
2267:
2248:
2236:
2216:
2210:
2191:
2070:
2064:
2055:
2049:
2037:
2025:
1967:
1945:
1936:
1924:
1921:
1906:
1897:
1865:
1850:
1838:
1829:
1722:
1703:
1700:
1681:
1657:
1651:
1639:
1633:
1621:
1612:
1476:
1448:
1421:
1415:
1403:
1394:
1315:
1296:
1290:
1271:
1226:
1200:
1194:
1175:
1148:
1142:
1130:
1124:
1106:
1094:
1065:
1059:
1047:
1041:
1023:
1011:
876:
816:
810:
759:
672:
654:
651:
636:
633:
621:
529:{\displaystyle x^{2}+xy+y^{2}}
383:Names of polynomials by degree
350:
337:
325:
312:
282:
269:
257:
244:
66:has been used as a synonym of
13:
1:
5047:Horner's method of evaluation
4672:
4541:Axler (1997) uses −∞. (p. 64)
3969:), the degree of the product
2801:The degree of the difference
2499:Degree of the zero polynomial
1596:is the sum of their degrees:
135:which can also be written as
4620:10.1007/978-0-387-89194-1_11
4588:, as long as deg 0 < deg
4558:{\displaystyle \mathbb {Z} }
77:For example, the polynomial
7:
5052:Polynomial identity testing
4768:Beyond the Quartic Equation
4722:Childs, Lindsay N. (2009),
4702:Childs, Lindsay N. (1995),
4231:
4213:(which each had degree 1).
3926:. In the special case that
3680:, and also a polynomial in
3214:{\displaystyle {\sqrt {x}}}
1674:For example, the degree of
1438:For example, the degree of
1168:For example, the degree of
990:
979:, with highest exponent 5.
749:, with highest exponent 3.
609:
599:{\displaystyle x^{2}+y^{2}}
479:, which are based on Latin
454:(or, less commonly, heptic)
10:
5104:
4378:{\displaystyle f(x)=a_{0}}
4261:For simplicity, this is a
3922:that have coefficients in
2915:The degree of the product
1555:{\displaystyle x^{2}+3x-2}
447:(or, less commonly, hexic)
5024:
4963:
4876:
4685:Linear Algebra Done Right
4429:
2838:{\displaystyle (x)-(x)=0}
2563:{\displaystyle -\infty ,}
2159:{\displaystyle Q=x^{2}-1}
2120:{\displaystyle P=x^{3}+x}
1583:Examples of vector spaces
4658:{\displaystyle -\infty }
4581:{\displaystyle -\infty }
4532:{\displaystyle -\infty }
4501:{\displaystyle -\infty }
4474:{\displaystyle -\infty }
4248:
3525:{\displaystyle dx^{d-1}}
3306:{\displaystyle \infty .}
3250:{\displaystyle \ \log x}
2985:{\displaystyle -\infty }
2861:{\displaystyle -\infty }
2523:{\displaystyle -\infty }
5037:Greatest common divisor
4145:, the ring of integers
3612:{\displaystyle x\log x}
4909:Quadratic function (2)
4659:
4582:
4559:
4533:
4502:
4475:
4414:
4379:
4281:Gullberg, Jan (1997),
4263:homogeneous polynomial
4139:
4097:
3940:principal ideal domain
3934:, the polynomial ring
3879:
3613:
3584:
3568:analysis of algorithms
3553:
3526:
3475:
3380:
3349:
3307:
3284:
3283:{\displaystyle \exp x}
3251:
3215:
3184:
3157:multiplicative inverse
3134:
3024:
2986:
2963:
2906:
2862:
2839:
2792:
2748:
2684:The degree of the sum
2667:
2614:
2564:
2524:
2489:
2421:
2392:
2369:
2328:
2160:
2121:
2080:
2006:
1986:
1958:
1878:
1806:
1764:
1664:
1556:
1514:
1428:
1360:
1255:
1158:
1075:
973:
883:
743:
679:
600:
560:, and (less commonly)
548:; the common ones are
540:due to two variables,
530:
372:
295:
224:
129:
4892:Constant function (0)
4818:Discourses on Algebra
4660:
4583:
4560:
4534:
4503:
4476:
4415:
4413:{\displaystyle a_{0}}
4380:
4140:
4098:
3880:
3672:, is a polynomial in
3614:
3585:
3554:
3552:{\displaystyle x^{d}}
3527:
3476:
3381:
3350:
3308:
3285:
3252:
3216:
3185:
3183:{\displaystyle \ 1/x}
3135:
3025:
2987:
2964:
2907:
2863:
2840:
2793:
2749:
2668:
2615:
2565:
2525:
2490:
2422:
2393:
2370:
2329:
2161:
2122:
2081:
2007:
1987:
1959:
1879:
1807:
1765:
1665:
1557:
1515:
1429:
1361:
1256:
1159:
1076:
974:
884:
744:
680:
601:
531:
373:
296:
225:
130:
60:univariate polynomial
5025:Tools and algorithms
4945:Quintic function (5)
4933:Quartic function (4)
4870:polynomial functions
4813:Shafarevich, Igor R.
4646:
4569:
4547:
4520:
4489:
4462:
4430:Lang, Serge (2005),
4397:
4347:
4238:Abel–Ruffini theorem
4114:
4000:
3695:
3594:
3574:
3536:
3500:
3400:
3379:{\displaystyle -1/2}
3359:
3320:
3294:
3268:
3262:exponential function
3232:
3201:
3163:
3054:
2996:
2973:
2919:
2872:
2849:
2805:
2758:
2688:
2630:
2577:
2548:
2511:
2431:
2420:{\displaystyle 1+2x}
2402:
2379:
2341:
2334:which has degree 6.
2170:
2131:
2092:
2016:
1996:
1976:
1888:
1820:
1781:
1678:
1603:
1524:
1442:
1385:
1268:
1172:
1085:
1002:
893:
756:
689:
618:
570:
491:
481:distributive numbers
402:Degree 0 – non-zero
309:
241:
139:
81:
27:Mathematical concept
4955:Septic equation (7)
4950:Sextic equation (6)
4897:Linear function (1)
3393:from its values is
3045:asymptotic analysis
4921:Cubic function (3)
4914:Quadratic equation
4786:Mac Lane, Saunders
4655:
4578:
4555:
4529:
4498:
4471:
4410:
4375:
4330:Mechanics Magazine
4197:+ 1 = 1. Thus deg(
4135:
4093:
3875:
3609:
3580:
3549:
3522:
3471:
3430:
3376:
3345:
3303:
3280:
3260:The degree of the
3247:
3224:The degree of the
3211:
3193:The degree of the
3180:
3155:The degree of the
3130:
3084:
3020:
2982:
2959:
2902:
2858:
2835:
2788:
2744:
2663:
2610:
2560:
2520:
2503:The degree of the
2485:
2417:
2391:{\displaystyle 2x}
2388:
2365:
2324:
2156:
2117:
2076:
2002:
1982:
1954:
1874:
1802:
1760:
1660:
1552:
1510:
1424:
1356:
1251:
1154:
1071:
969:
879:
739:
675:
596:
526:
368:
291:
220:
125:
5075:
5074:
5016:Quasi-homogeneous
4790:Birkhoff, Garrett
4441:978-0-387-95385-4
4332:, Vol. LV, p. 171
3688:. The polynomial
3583:{\displaystyle x}
3469:
3415:
3343:
3337:
3237:
3209:
3168:
3128:
3069:
2542:negative infinity
2127:has degree 3 and
2005:{\displaystyle Q}
1985:{\displaystyle P}
1814:integers modulo 4
463:Degree 10 – decic
16:(Redirected from
5095:
4938:Quartic equation
4859:
4852:
4845:
4836:
4835:
4831:
4808:
4781:
4758:
4745:Abstract Algebra
4738:
4718:
4698:
4666:
4664:
4662:
4661:
4656:
4640:
4634:
4632:
4607:
4601:
4587:
4585:
4584:
4579:
4564:
4562:
4561:
4556:
4554:
4538:
4536:
4535:
4530:
4507:
4505:
4504:
4499:
4480:
4478:
4477:
4472:
4451:
4445:
4444:
4427:
4421:
4419:
4417:
4416:
4411:
4409:
4408:
4384:
4382:
4381:
4376:
4374:
4373:
4341:
4335:
4322:
4316:
4313:
4307:
4304:
4298:
4297:
4278:
4266:
4259:
4144:
4142:
4141:
4136:
4134:
4126:
4121:
4102:
4100:
4099:
4094:
3944:Euclidean domain
3892:and degree 2 in
3888:has degree 3 in
3884:
3882:
3881:
3876:
3871:
3870:
3837:
3836:
3824:
3823:
3793:
3792:
3780:
3779:
3764:
3763:
3733:
3732:
3717:
3716:
3707:
3706:
3618:
3616:
3615:
3610:
3589:
3587:
3586:
3581:
3558:
3556:
3555:
3550:
3548:
3547:
3531:
3529:
3528:
3523:
3521:
3520:
3487:L'Hôpital's rule
3480:
3478:
3477:
3472:
3470:
3468:
3454:
3444:
3432:
3429:
3385:
3383:
3382:
3377:
3372:
3354:
3352:
3351:
3346:
3344:
3339:
3338:
3333:
3324:
3312:
3310:
3309:
3304:
3289:
3287:
3286:
3281:
3256:
3254:
3253:
3248:
3235:
3220:
3218:
3217:
3212:
3210:
3205:
3189:
3187:
3186:
3181:
3176:
3166:
3139:
3137:
3136:
3131:
3129:
3127:
3116:
3115:
3098:
3086:
3083:
3029:
3027:
3026:
3021:
2991:
2989:
2988:
2983:
2968:
2966:
2965:
2960:
2943:
2942:
2911:
2909:
2908:
2903:
2867:
2865:
2864:
2859:
2844:
2842:
2841:
2836:
2797:
2795:
2794:
2789:
2753:
2751:
2750:
2745:
2737:
2736:
2703:
2702:
2672:
2670:
2669:
2664:
2619:
2617:
2616:
2611:
2569:
2567:
2566:
2561:
2529:
2527:
2526:
2521:
2494:
2492:
2491:
2486:
2426:
2424:
2423:
2418:
2397:
2395:
2394:
2389:
2374:
2372:
2371:
2366:
2361:
2353:
2348:
2333:
2331:
2330:
2325:
2314:
2313:
2298:
2297:
2282:
2281:
2260:
2259:
2244:
2243:
2228:
2227:
2203:
2202:
2165:
2163:
2162:
2157:
2149:
2148:
2126:
2124:
2123:
2118:
2110:
2109:
2088:For example, if
2085:
2083:
2082:
2077:
2011:
2009:
2008:
2003:
1991:
1989:
1988:
1983:
1963:
1961:
1960:
1955:
1883:
1881:
1880:
1875:
1811:
1809:
1808:
1803:
1801:
1793:
1788:
1769:
1767:
1766:
1761:
1753:
1752:
1737:
1736:
1715:
1714:
1693:
1692:
1669:
1667:
1666:
1661:
1581:; for more, see
1561:
1559:
1558:
1553:
1536:
1535:
1519:
1517:
1516:
1511:
1494:
1493:
1460:
1459:
1433:
1431:
1430:
1425:
1365:
1363:
1362:
1357:
1343:
1342:
1330:
1329:
1308:
1307:
1283:
1282:
1260:
1258:
1257:
1252:
1244:
1243:
1225:
1224:
1212:
1211:
1187:
1186:
1163:
1161:
1160:
1155:
1080:
1078:
1077:
1072:
978:
976:
975:
970:
953:
952:
937:
936:
921:
920:
905:
904:
888:
886:
885:
880:
866:
865:
850:
849:
834:
833:
803:
802:
787:
786:
774:
773:
748:
746:
745:
740:
723:
722:
707:
706:
684:
682:
681:
676:
605:
603:
602:
597:
595:
594:
582:
581:
535:
533:
532:
527:
525:
524:
503:
502:
460:Degree 9 – nonic
457:Degree 8 – octic
377:
375:
374:
369:
358:
357:
333:
332:
300:
298:
297:
292:
290:
289:
265:
264:
229:
227:
226:
221:
216:
215:
206:
205:
190:
189:
180:
179:
164:
163:
154:
153:
134:
132:
131:
126:
106:
105:
96:
95:
48:degree of a term
21:
5103:
5102:
5098:
5097:
5096:
5094:
5093:
5092:
5078:
5077:
5076:
5071:
5020:
4959:
4902:Linear equation
4872:
4863:
4829:
4806:
4779:
4756:
4736:
4716:
4696:
4675:
4670:
4669:
4647:
4644:
4643:
4641:
4637:
4630:
4608:
4604:
4570:
4567:
4566:
4550:
4548:
4545:
4544:
4542:
4540:
4521:
4518:
4517:
4512:any integer or
4490:
4487:
4486:
4463:
4460:
4459:
4456:
4454:
4452:
4448:
4442:
4428:
4424:
4404:
4400:
4398:
4395:
4394:
4369:
4365:
4348:
4345:
4344:
4342:
4338:
4323:
4319:
4314:
4310:
4305:
4301:
4295:
4279:
4275:
4270:
4269:
4260:
4256:
4251:
4234:
4151:integral domain
4130:
4122:
4117:
4115:
4112:
4111:
4001:
3998:
3997:
3913:polynomial ring
3902:
3866:
3862:
3832:
3828:
3819:
3815:
3788:
3784:
3775:
3771:
3759:
3755:
3728:
3724:
3712:
3708:
3702:
3698:
3696:
3693:
3692:
3629:
3595:
3592:
3591:
3575:
3572:
3571:
3543:
3539:
3537:
3534:
3533:
3510:
3506:
3501:
3498:
3497:
3455:
3437:
3433:
3431:
3419:
3401:
3398:
3397:
3368:
3360:
3357:
3356:
3332:
3325:
3323:
3321:
3318:
3317:
3295:
3292:
3291:
3269:
3266:
3265:
3233:
3230:
3229:
3204:
3202:
3199:
3198:
3172:
3164:
3161:
3160:
3117:
3111:
3094:
3087:
3085:
3073:
3055:
3052:
3051:
3043:. One based on
3037:
2997:
2994:
2993:
2974:
2971:
2970:
2938:
2934:
2920:
2917:
2916:
2873:
2870:
2869:
2850:
2847:
2846:
2806:
2803:
2802:
2759:
2756:
2755:
2732:
2728:
2698:
2694:
2689:
2686:
2685:
2631:
2628:
2627:
2578:
2575:
2574:
2549:
2546:
2545:
2535:zero polynomial
2512:
2509:
2508:
2505:zero polynomial
2501:
2432:
2429:
2428:
2403:
2400:
2399:
2380:
2377:
2376:
2357:
2349:
2344:
2342:
2339:
2338:
2309:
2305:
2293:
2289:
2277:
2273:
2255:
2251:
2239:
2235:
2223:
2219:
2198:
2194:
2171:
2168:
2167:
2144:
2140:
2132:
2129:
2128:
2105:
2101:
2093:
2090:
2089:
2017:
2014:
2013:
1997:
1994:
1993:
1977:
1974:
1973:
1970:
1889:
1886:
1885:
1821:
1818:
1817:
1816:, one has that
1797:
1789:
1784:
1782:
1779:
1778:
1748:
1744:
1732:
1728:
1710:
1706:
1688:
1684:
1679:
1676:
1675:
1604:
1601:
1600:
1594:integral domain
1531:
1527:
1525:
1522:
1521:
1489:
1485:
1455:
1451:
1443:
1440:
1439:
1386:
1383:
1382:
1372:
1338:
1334:
1325:
1321:
1303:
1299:
1278:
1274:
1269:
1266:
1265:
1239:
1235:
1220:
1216:
1207:
1203:
1182:
1178:
1173:
1170:
1169:
1086:
1083:
1082:
1003:
1000:
999:
993:
985:
948:
944:
932:
928:
916:
912:
900:
896:
894:
891:
890:
861:
857:
845:
841:
829:
825:
798:
794:
782:
778:
769:
765:
757:
754:
753:
752:The polynomial
718:
714:
702:
698:
690:
687:
686:
619:
616:
615:
614:The polynomial
612:
590:
586:
577:
573:
571:
568:
567:
520:
516:
498:
494:
492:
489:
488:
469:ordinal numbers
391:Special case –
385:
353:
349:
328:
324:
310:
307:
306:
285:
281:
260:
256:
242:
239:
238:
211:
207:
201:
197:
185:
181:
175:
171:
159:
155:
149:
145:
140:
137:
136:
101:
97:
91:
87:
82:
79:
78:
28:
23:
22:
15:
12:
11:
5:
5101:
5091:
5090:
5073:
5072:
5070:
5069:
5064:
5059:
5054:
5049:
5044:
5039:
5034:
5028:
5026:
5022:
5021:
5019:
5018:
5013:
5008:
5003:
4998:
4993:
4988:
4983:
4978:
4973:
4967:
4965:
4961:
4960:
4958:
4957:
4952:
4947:
4942:
4941:
4940:
4930:
4929:
4928:
4926:Cubic equation
4918:
4917:
4916:
4906:
4905:
4904:
4894:
4889:
4883:
4881:
4874:
4873:
4862:
4861:
4854:
4847:
4839:
4833:
4832:
4827:
4809:
4804:
4782:
4777:
4763:King, R. Bruce
4759:
4754:
4739:
4734:
4719:
4714:
4699:
4694:
4680:Axler, Sheldon
4674:
4671:
4668:
4667:
4654:
4651:
4635:
4628:
4602:
4577:
4574:
4553:
4528:
4525:
4497:
4494:
4470:
4467:
4446:
4440:
4422:
4407:
4403:
4372:
4368:
4364:
4361:
4358:
4355:
4352:
4336:
4317:
4308:
4299:
4293:
4272:
4271:
4268:
4267:
4253:
4252:
4250:
4247:
4246:
4245:
4240:
4233:
4230:
4133:
4129:
4125:
4120:
4104:
4103:
4092:
4089:
4086:
4083:
4080:
4077:
4074:
4071:
4068:
4065:
4062:
4059:
4056:
4053:
4050:
4047:
4044:
4041:
4038:
4035:
4032:
4029:
4026:
4023:
4020:
4017:
4014:
4011:
4008:
4005:
3901:
3898:
3886:
3885:
3874:
3869:
3865:
3861:
3858:
3855:
3852:
3849:
3846:
3843:
3840:
3835:
3831:
3827:
3822:
3818:
3814:
3811:
3808:
3805:
3802:
3799:
3796:
3791:
3787:
3783:
3778:
3774:
3770:
3767:
3762:
3758:
3754:
3751:
3748:
3745:
3742:
3739:
3736:
3731:
3727:
3723:
3720:
3715:
3711:
3705:
3701:
3628:
3625:
3608:
3605:
3602:
3599:
3579:
3564:big O notation
3546:
3542:
3519:
3516:
3513:
3509:
3505:
3483:
3482:
3467:
3464:
3461:
3458:
3453:
3450:
3447:
3443:
3440:
3436:
3428:
3425:
3422:
3418:
3414:
3411:
3408:
3405:
3375:
3371:
3367:
3364:
3342:
3336:
3331:
3328:
3314:
3313:
3302:
3299:
3279:
3276:
3273:
3258:
3246:
3243:
3240:
3222:
3208:
3191:
3190:, is −1.
3179:
3175:
3171:
3142:
3141:
3126:
3123:
3120:
3114:
3110:
3107:
3104:
3101:
3097:
3093:
3090:
3082:
3079:
3076:
3072:
3068:
3065:
3062:
3059:
3036:
3033:
3032:
3031:
3019:
3016:
3013:
3010:
3007:
3004:
3001:
2981:
2978:
2958:
2955:
2952:
2949:
2946:
2941:
2937:
2933:
2930:
2927:
2924:
2913:
2901:
2898:
2895:
2892:
2889:
2886:
2883:
2880:
2877:
2857:
2854:
2834:
2831:
2828:
2825:
2822:
2819:
2816:
2813:
2810:
2799:
2787:
2784:
2781:
2778:
2775:
2772:
2769:
2766:
2763:
2743:
2740:
2735:
2731:
2727:
2724:
2721:
2718:
2715:
2712:
2709:
2706:
2701:
2697:
2693:
2678:behavior rules
2674:
2673:
2662:
2659:
2656:
2653:
2650:
2647:
2644:
2641:
2638:
2635:
2621:
2620:
2609:
2606:
2603:
2600:
2597:
2594:
2591:
2588:
2585:
2582:
2559:
2556:
2553:
2519:
2516:
2500:
2497:
2484:
2481:
2478:
2475:
2472:
2469:
2466:
2463:
2460:
2457:
2454:
2451:
2448:
2445:
2442:
2439:
2436:
2416:
2413:
2410:
2407:
2387:
2384:
2364:
2360:
2356:
2352:
2347:
2323:
2320:
2317:
2312:
2308:
2304:
2301:
2296:
2292:
2288:
2285:
2280:
2276:
2272:
2269:
2266:
2263:
2258:
2254:
2250:
2247:
2242:
2238:
2234:
2231:
2226:
2222:
2218:
2215:
2212:
2209:
2206:
2201:
2197:
2193:
2190:
2187:
2184:
2181:
2178:
2175:
2155:
2152:
2147:
2143:
2139:
2136:
2116:
2113:
2108:
2104:
2100:
2097:
2075:
2072:
2069:
2066:
2063:
2060:
2057:
2054:
2051:
2048:
2045:
2042:
2039:
2036:
2033:
2030:
2027:
2024:
2021:
2001:
1981:
1969:
1966:
1953:
1950:
1947:
1944:
1941:
1938:
1935:
1932:
1929:
1926:
1923:
1920:
1917:
1914:
1911:
1908:
1905:
1902:
1899:
1896:
1893:
1873:
1870:
1867:
1864:
1861:
1858:
1855:
1852:
1849:
1846:
1843:
1840:
1837:
1834:
1831:
1828:
1825:
1800:
1796:
1792:
1787:
1759:
1756:
1751:
1747:
1743:
1740:
1735:
1731:
1727:
1724:
1721:
1718:
1713:
1709:
1705:
1702:
1699:
1696:
1691:
1687:
1683:
1672:
1671:
1659:
1656:
1653:
1650:
1647:
1644:
1641:
1638:
1635:
1632:
1629:
1626:
1623:
1620:
1617:
1614:
1611:
1608:
1551:
1548:
1545:
1542:
1539:
1534:
1530:
1509:
1506:
1503:
1500:
1497:
1492:
1488:
1484:
1481:
1478:
1475:
1472:
1469:
1466:
1463:
1458:
1454:
1450:
1447:
1436:
1435:
1423:
1420:
1417:
1414:
1411:
1408:
1405:
1402:
1399:
1396:
1393:
1390:
1371:
1370:Multiplication
1368:
1355:
1352:
1349:
1346:
1341:
1337:
1333:
1328:
1324:
1320:
1317:
1314:
1311:
1306:
1302:
1298:
1295:
1292:
1289:
1286:
1281:
1277:
1273:
1250:
1247:
1242:
1238:
1234:
1231:
1228:
1223:
1219:
1215:
1210:
1206:
1202:
1199:
1196:
1193:
1190:
1185:
1181:
1177:
1166:
1165:
1153:
1150:
1147:
1144:
1141:
1138:
1135:
1132:
1129:
1126:
1123:
1120:
1117:
1114:
1111:
1108:
1105:
1102:
1099:
1096:
1093:
1090:
1070:
1067:
1064:
1061:
1058:
1055:
1052:
1049:
1046:
1043:
1040:
1037:
1034:
1031:
1028:
1025:
1022:
1019:
1016:
1013:
1010:
1007:
992:
989:
984:
981:
968:
965:
962:
959:
956:
951:
947:
943:
940:
935:
931:
927:
924:
919:
915:
911:
908:
903:
899:
878:
875:
872:
869:
864:
860:
856:
853:
848:
844:
840:
837:
832:
828:
824:
821:
818:
815:
812:
809:
806:
801:
797:
793:
790:
785:
781:
777:
772:
768:
764:
761:
738:
735:
732:
729:
726:
721:
717:
713:
710:
705:
701:
697:
694:
674:
671:
668:
665:
662:
659:
656:
653:
650:
647:
644:
641:
638:
635:
632:
629:
626:
623:
611:
608:
593:
589:
585:
580:
576:
523:
519:
515:
512:
509:
506:
501:
497:
465:
464:
461:
458:
455:
448:
441:
435:
424:
418:
412:
406:
400:
384:
381:
367:
364:
361:
356:
352:
348:
345:
342:
339:
336:
331:
327:
323:
320:
317:
314:
303:distributivity
288:
284:
280:
277:
274:
271:
268:
263:
259:
255:
252:
249:
246:
219:
214:
210:
204:
200:
196:
193:
188:
184:
178:
174:
170:
167:
162:
158:
152:
148:
144:
124:
121:
118:
115:
112:
109:
104:
100:
94:
90:
86:
26:
9:
6:
4:
3:
2:
5100:
5089:
5086:
5085:
5083:
5068:
5067:Gröbner basis
5065:
5063:
5060:
5058:
5055:
5053:
5050:
5048:
5045:
5043:
5040:
5038:
5035:
5033:
5032:Factorization
5030:
5029:
5027:
5023:
5017:
5014:
5012:
5009:
5007:
5004:
5002:
4999:
4997:
4994:
4992:
4989:
4987:
4984:
4982:
4979:
4977:
4974:
4972:
4969:
4968:
4966:
4964:By properties
4962:
4956:
4953:
4951:
4948:
4946:
4943:
4939:
4936:
4935:
4934:
4931:
4927:
4924:
4923:
4922:
4919:
4915:
4912:
4911:
4910:
4907:
4903:
4900:
4899:
4898:
4895:
4893:
4890:
4888:
4885:
4884:
4882:
4880:
4875:
4871:
4867:
4860:
4855:
4853:
4848:
4846:
4841:
4840:
4837:
4830:
4828:9783540422532
4824:
4820:
4819:
4814:
4810:
4807:
4805:9780821816462
4801:
4797:
4796:
4791:
4787:
4783:
4780:
4778:9780817648497
4774:
4770:
4769:
4764:
4760:
4757:
4755:9780387715681
4751:
4747:
4746:
4740:
4737:
4735:9780387745275
4731:
4727:
4726:
4720:
4717:
4715:9780387989990
4711:
4707:
4706:
4700:
4697:
4695:9780387982595
4691:
4687:
4686:
4681:
4677:
4676:
4649:
4639:
4631:
4629:9780387891941
4625:
4621:
4617:
4613:
4606:
4599:
4595:
4591:
4572:
4523:
4515:
4511:
4492:
4484:
4465:
4450:
4443:
4437:
4433:
4426:
4405:
4401:
4392:
4388:
4370:
4366:
4362:
4356:
4350:
4340:
4333:
4331:
4326:
4321:
4312:
4303:
4296:
4294:9780393040029
4290:
4286:
4285:
4277:
4273:
4264:
4258:
4254:
4244:
4241:
4239:
4236:
4235:
4229:
4227:
4223:
4219:
4214:
4212:
4208:
4204:
4200:
4196:
4192:
4188:
4184:
4180:
4176:
4172:
4168:
4164:
4160:
4156:
4152:
4148:
4127:
4123:
4109:
4084:
4078:
4072:
4069:
4066:
4057:
4051:
4045:
4042:
4039:
4030:
4024:
4018:
4012:
4006:
4003:
3996:
3995:
3994:
3992:
3988:
3984:
3980:
3976:
3972:
3968:
3964:
3960:
3956:
3952:
3947:
3945:
3941:
3937:
3933:
3929:
3925:
3921:
3917:
3914:
3910:
3907:
3897:
3895:
3891:
3867:
3863:
3859:
3853:
3850:
3844:
3838:
3833:
3829:
3820:
3816:
3809:
3803:
3800:
3794:
3789:
3785:
3776:
3772:
3765:
3760:
3756:
3749:
3743:
3740:
3737:
3734:
3729:
3725:
3721:
3718:
3713:
3709:
3703:
3699:
3691:
3690:
3689:
3687:
3683:
3679:
3675:
3671:
3667:
3662:
3660:
3657:
3653:
3649:
3645:
3642:
3638:
3634:
3624:
3622:
3606:
3603:
3600:
3597:
3577:
3569:
3565:
3560:
3544:
3540:
3517:
3514:
3511:
3507:
3503:
3496:
3492:
3488:
3462:
3456:
3448:
3441:
3438:
3434:
3420:
3412:
3409:
3406:
3403:
3396:
3395:
3394:
3392:
3387:
3373:
3369:
3365:
3362:
3340:
3334:
3329:
3326:
3300:
3277:
3274:
3271:
3263:
3259:
3244:
3241:
3238:
3227:
3223:
3206:
3196:
3192:
3177:
3173:
3169:
3158:
3154:
3153:
3152:
3149:
3147:
3124:
3121:
3118:
3105:
3099:
3091:
3088:
3074:
3066:
3063:
3060:
3057:
3050:
3049:
3048:
3046:
3042:
3017:
3014:
3008:
3005:
2999:
2976:
2956:
2953:
2947:
2944:
2939:
2935:
2925:
2914:
2896:
2893:
2890:
2881:
2875:
2852:
2832:
2829:
2823:
2817:
2811:
2800:
2779:
2776:
2773:
2764:
2761:
2741:
2738:
2733:
2729:
2725:
2719:
2713:
2707:
2704:
2699:
2695:
2683:
2682:
2681:
2679:
2660:
2654:
2651:
2642:
2636:
2633:
2626:
2625:
2624:
2607:
2604:
2601:
2592:
2589:
2586:
2573:
2572:
2571:
2557:
2551:
2543:
2538:
2536:
2531:
2514:
2506:
2496:
2495:of degree 0.
2482:
2479:
2476:
2473:
2470:
2467:
2464:
2461:
2455:
2452:
2449:
2446:
2440:
2437:
2434:
2414:
2411:
2408:
2405:
2385:
2382:
2362:
2354:
2350:
2335:
2321:
2318:
2315:
2310:
2306:
2302:
2299:
2294:
2290:
2286:
2283:
2278:
2274:
2270:
2264:
2261:
2256:
2252:
2245:
2240:
2232:
2229:
2224:
2220:
2213:
2207:
2204:
2199:
2195:
2188:
2185:
2182:
2179:
2176:
2173:
2153:
2150:
2145:
2141:
2137:
2134:
2114:
2111:
2106:
2102:
2098:
2095:
2086:
2073:
2067:
2061:
2058:
2052:
2046:
2043:
2040:
2034:
2031:
2028:
2022:
2019:
1999:
1979:
1965:
1951:
1948:
1942:
1939:
1933:
1930:
1927:
1918:
1915:
1912:
1909:
1903:
1900:
1894:
1891:
1871:
1868:
1862:
1859:
1856:
1853:
1847:
1844:
1841:
1835:
1832:
1826:
1823:
1815:
1794:
1790:
1776:
1771:
1757:
1754:
1749:
1745:
1741:
1738:
1733:
1729:
1725:
1719:
1716:
1711:
1707:
1697:
1694:
1689:
1685:
1654:
1648:
1645:
1642:
1636:
1630:
1627:
1624:
1618:
1615:
1609:
1606:
1599:
1598:
1597:
1595:
1591:
1586:
1584:
1580:
1576:
1572:
1568:
1563:
1549:
1546:
1543:
1540:
1537:
1532:
1528:
1507:
1504:
1501:
1498:
1495:
1490:
1486:
1482:
1479:
1473:
1470:
1467:
1464:
1461:
1456:
1452:
1445:
1418:
1412:
1409:
1406:
1400:
1397:
1391:
1388:
1381:
1380:
1379:
1377:
1367:
1353:
1350:
1347:
1344:
1339:
1335:
1331:
1326:
1322:
1318:
1312:
1309:
1304:
1300:
1293:
1287:
1284:
1279:
1275:
1262:
1248:
1245:
1240:
1236:
1232:
1229:
1221:
1217:
1213:
1208:
1204:
1197:
1191:
1188:
1183:
1179:
1145:
1139:
1136:
1133:
1127:
1121:
1118:
1109:
1103:
1100:
1097:
1091:
1088:
1062:
1056:
1053:
1050:
1044:
1038:
1035:
1026:
1020:
1017:
1014:
1008:
1005:
998:
997:
996:
988:
980:
966:
963:
960:
957:
954:
949:
945:
941:
938:
933:
929:
925:
922:
917:
913:
909:
906:
901:
897:
873:
870:
867:
862:
858:
854:
851:
846:
842:
838:
835:
830:
826:
822:
819:
813:
807:
804:
799:
795:
791:
788:
783:
779:
775:
770:
766:
762:
750:
736:
733:
730:
727:
724:
719:
715:
711:
708:
703:
699:
695:
692:
669:
666:
663:
660:
657:
648:
645:
642:
639:
630:
627:
624:
607:
591:
587:
583:
578:
574:
565:
564:
559:
558:
553:
552:
547:
543:
539:
521:
517:
513:
510:
507:
504:
499:
495:
486:
483:, and end in
482:
478:
474:
471:, and end in
470:
462:
459:
456:
453:
449:
446:
442:
440:
436:
433:
429:
425:
423:
419:
417:
413:
411:
407:
405:
401:
398:
394:
390:
389:
388:
380:
365:
362:
359:
354:
346:
343:
340:
334:
329:
321:
318:
315:
304:
286:
278:
275:
272:
266:
261:
253:
250:
247:
235:
233:
217:
212:
208:
202:
198:
194:
191:
186:
182:
176:
172:
168:
165:
160:
156:
150:
146:
142:
122:
119:
116:
113:
110:
107:
102:
98:
92:
88:
84:
75:
73:
69:
65:
61:
57:
53:
49:
45:
41:
37:
33:
19:
5062:Discriminant
4981:Multivariate
4878:
4817:
4794:
4767:
4744:
4724:
4704:
4684:
4638:
4611:
4605:
4597:
4593:
4589:
4513:
4509:
4482:
4449:
4431:
4425:
4390:
4386:
4339:
4329:
4325:James Cockle
4320:
4311:
4302:
4283:
4276:
4257:
4225:
4221:
4217:
4215:
4210:
4206:
4202:
4198:
4194:
4190:
4186:
4182:
4178:
4174:
4170:
4166:
4162:
4158:
4154:
4107:
4105:
3990:
3986:
3982:
3978:
3974:
3970:
3966:
3962:
3958:
3954:
3950:
3948:
3935:
3927:
3923:
3919:
3915:
3908:
3903:
3893:
3889:
3887:
3685:
3681:
3677:
3673:
3669:
3665:
3663:
3658:
3655:
3651:
3647:
3643:
3640:
3637:total degree
3636:
3632:
3630:
3620:
3561:
3490:
3484:
3390:
3388:
3315:
3150:
3146:log–log plot
3143:
3040:
3038:
2675:
2622:
2541:
2539:
2532:
2502:
2336:
2087:
1971:
1772:
1673:
1587:
1579:vector space
1574:
1570:
1564:
1437:
1373:
1263:
1167:
994:
986:
751:
613:
561:
555:
549:
545:
541:
537:
484:
472:
466:
386:
236:
76:
67:
63:
35:
29:
18:Total degree
5088:Polynomials
5011:Homogeneous
5006:Square-free
5001:Irreducible
4866:Polynomials
4173:+ 1. Then,
3195:square root
1968:Composition
450:Degree 7 –
443:Degree 6 –
437:Degree 5 –
432:biquadratic
426:Degree 4 –
420:Degree 3 –
414:Degree 2 –
408:Degree 1 –
32:mathematics
4971:Univariate
4673:References
4420:." (p. 23)
4216:Since the
3930:is also a
3495:derivative
1565:Thus, the
40:polynomial
5057:Resultant
4996:Trinomial
4976:Bivariate
4653:∞
4650:−
4576:∞
4573:−
4527:∞
4524:−
4496:∞
4493:−
4469:∞
4466:−
4073:
4046:
4007:
3604:
3566:. In the
3515:−
3427:∞
3424:→
3407:
3363:−
3298:∞
3275:
3242:
3226:logarithm
3221:, is 1/2.
3122:
3092:
3081:∞
3078:→
3061:
3012:∞
3009:−
3003:∞
3000:−
2980:∞
2977:−
2882:≤
2879:∞
2876:−
2856:∞
2853:−
2818:−
2783:∞
2780:−
2765:≤
2658:∞
2655:−
2646:∞
2643:−
2596:∞
2593:−
2555:∞
2552:−
2518:∞
2515:−
2441:∘
2316:−
2284:−
2262:−
2230:−
2205:−
2189:∘
2177:∘
2151:−
2062:
2047:
2032:∘
2023:
1934:
1895:
1848:
1827:
1649:
1631:
1610:
1547:−
1505:−
1471:−
1413:
1392:
1233:−
1198:−
1140:
1122:
1110:≤
1101:−
1092:
1057:
1039:
1027:≤
1009:
939:−
820:−
789:−
709:−
693:−
667:−
658:−
628:−
563:trinomial
542:quadratic
416:quadratic
344:−
335:−
276:−
267:−
192:−
117:−
52:variables
44:monomials
5082:Category
5042:Division
4991:Binomial
4986:Monomial
4815:(2003),
4792:(1999),
4765:(2009),
4682:(1997),
4592:for all
4387:constant
4232:See also
3904:Given a
3442:′
1577:forms a
991:Addition
610:Examples
557:binomial
551:monomial
404:constant
399:, below)
58:. For a
4795:Algebra
4596:≠ 0." (
4432:Algebra
3257:, is 0.
2680:above:
566:; thus
546:-nomial
439:quintic
428:quartic
56:integer
4879:degree
4825:
4802:
4775:
4752:
4732:
4712:
4692:
4626:
4565:or as
4438:
4291:
4147:modulo
3961:) and
3911:, the
3236:
3167:
1884:, but
1592:or an
1376:scalar
538:binary
452:septic
445:sextic
410:linear
232:powers
68:degree
36:degree
34:, the
4249:Notes
4189:) = 4
4169:) = 2
3938:is a
3932:field
3290:, is
1590:field
477:arity
422:cubic
395:(see
64:order
38:of a
4868:and
4823:ISBN
4800:ISBN
4773:ISBN
4750:ISBN
4730:ISBN
4710:ISBN
4690:ISBN
4624:ISBN
4508:for
4436:ISBN
4289:ISBN
4218:norm
4209:and
4161:) =
3989:and
3951:norm
3906:ring
3668:and
3621:same
3590:and
2623:and
2398:and
1992:and
1775:ring
1081:and
485:-ary
393:zero
4877:By
4616:doi
4193:+ 4
4070:deg
4043:deg
4004:deg
3650:+ 4
3646:+ 3
3633:sum
3601:log
3532:of
3417:lim
3404:deg
3355:is
3272:exp
3239:log
3119:log
3089:log
3071:lim
3058:deg
3047:is
2969:is
2885:max
2845:is
2768:max
2581:max
2530:).
2059:deg
2044:deg
2020:deg
1931:deg
1892:deg
1845:deg
1824:deg
1812:of
1646:deg
1628:deg
1607:deg
1585:.
1567:set
1410:deg
1389:deg
1137:deg
1119:deg
1113:max
1089:deg
1054:deg
1036:deg
1030:max
1006:deg
737:378
473:-ic
74:).
30:In
5084::
4788:;
4622:,
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3896:.
3661:.
3559:.
3386:.
3264:,
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3197:,
3159:,
3148:.
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1562:.
958:14
871:14
728:72
712:42
670:21
554:,
4858:e
4851:t
4844:v
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4598:A
4594:A
4590:A
4552:Z
4514:m
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4483:m
4406:0
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4354:(
4351:f
4334:)
4226:x
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4222:f
4211:g
4207:f
4203:g
4201:⋅
4199:f
4195:x
4191:x
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4185:(
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4181:)
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4177:(
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4124:/
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4028:(
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3991:g
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3868:3
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3860:3
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3807:)
3804:y
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3798:(
3795:+
3790:2
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3777:2
3773:y
3769:(
3766:+
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3750:3
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3741:y
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3735:+
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2912:.
2900:)
2897:1
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2720:0
2717:(
2714:+
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2708:x
2705:+
2700:3
2696:x
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2661:.
2652:=
2649:)
2640:(
2637:+
2634:a
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2605:a
2602:=
2599:)
2590:,
2587:a
2584:(
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2480:2
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2474:x
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2456:x
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2447:1
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2415:x
2412:2
2409:+
2406:1
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2359:Z
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2351:/
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2322:,
2319:2
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2307:x
2303:4
2300:+
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2287:3
2279:6
2275:x
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2268:)
2265:1
2257:2
2253:x
2249:(
2246:+
2241:3
2237:)
2233:1
2225:2
2221:x
2217:(
2214:=
2211:)
2208:1
2200:2
2196:x
2192:(
2186:P
2183:=
2180:Q
2174:P
2154:1
2146:2
2142:x
2138:=
2135:Q
2115:x
2112:+
2107:3
2103:x
2099:=
2096:P
2074:.
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2068:Q
2065:(
2056:)
2053:P
2050:(
2041:=
2038:)
2035:Q
2029:P
2026:(
2000:Q
1980:P
1952:1
1949:=
1946:)
1943:x
1940:2
1937:(
1928:=
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1922:)
1919:x
1916:2
1913:+
1910:1
1907:(
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1901:2
1898:(
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1863:x
1860:2
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1836:x
1833:2
1830:(
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1795:4
1791:/
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1755:+
1750:3
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1720:1
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1712:2
1708:x
1704:(
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1698:x
1695:+
1690:3
1686:x
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1670:.
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1655:Q
1652:(
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1637:P
1634:(
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1619:Q
1616:P
1613:(
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1538:+
1533:2
1529:x
1508:4
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1496:+
1491:2
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1474:2
1468:x
1465:3
1462:+
1457:2
1453:x
1449:(
1446:2
1434:.
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1419:P
1416:(
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1401:P
1398:c
1395:(
1354:1
1351:+
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1345:+
1340:2
1336:x
1332:+
1327:3
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1313:1
1310:+
1305:2
1301:x
1297:(
1294:+
1291:)
1288:x
1285:+
1280:3
1276:x
1272:(
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1222:2
1218:x
1214:+
1209:3
1205:x
1201:(
1195:)
1192:x
1189:+
1184:3
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1128:P
1125:(
1116:{
1107:)
1104:Q
1098:P
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1060:(
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1042:(
1033:{
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1021:Q
1018:+
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1012:(
967:6
964:+
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955:+
950:2
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926:2
923:+
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914:z
910:8
907:+
902:5
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877:)
874:z
868:+
863:3
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855:2
852:+
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817:(
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796:z
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776:+
771:8
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763:3
760:(
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725:+
720:2
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664:y
661:4
655:(
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649:6
646:+
643:y
640:2
637:(
634:)
631:3
625:y
622:(
592:2
588:y
584:+
579:2
575:x
522:2
518:y
514:+
511:y
508:x
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500:2
496:x
434:)
366:x
363:4
360:=
355:2
351:)
347:1
341:x
338:(
330:2
326:)
322:1
319:+
316:x
313:(
287:2
283:)
279:1
273:x
270:(
262:2
258:)
254:1
251:+
248:x
245:(
218:,
213:0
209:y
203:0
199:x
195:9
187:0
183:y
177:1
173:x
169:4
166:+
161:3
157:y
151:2
147:x
143:7
123:,
120:9
114:x
111:4
108:+
103:3
99:y
93:2
89:x
85:7
20:)
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