2601:
1701:
1367:
3681:
2410:
1065:
2425:
1378:
2198:
1076:
3391:
2419:= 3). Place the result (+3) below the bar. 3x has been divided leaving no remainder, and can therefore be marked as used. The result 3 is then multiplied by the second term in the divisor −3 = −9. Determine the partial remainder by subtracting −4 − (−9) = 5. Mark −4 as used and place the new remainder 5 above it.
2253:
755:
800:
2596:{\displaystyle {\begin{matrix}\quad \qquad \qquad \qquad {\bcancel {x^{2}}}\quad {\bcancel {3x}}\quad 5\\\qquad \quad {\bcancel {x^{3}}}+{\bcancel {-2x^{2}}}+{\bcancel {0x}}{\bcancel {-4}}\\{\underline {\div \qquad \qquad \qquad \qquad \qquad x-3}}\\x^{2}+x+3\qquad \end{matrix}}}
2006:
1696:{\displaystyle {\begin{array}{r}x^{2}+{\color {White}1}x+3\\x-3\ {\overline {)\ x^{3}-2x^{2}+0x-4}}\\{\underline {x^{3}-3x^{2}{\color {White}{}+0x-4}}}\\+x^{2}+0x{\color {White}{}-4}\\{\underline {+x^{2}-3x{\color {White}{}-4}}}\\+3x-4\\{\underline {+3x-9}}\\+5\end{array}}}
2061:
1362:{\displaystyle {\begin{array}{r}x^{2}+{\color {White}1}x{\color {White}{}+3}\\x-3\ {\overline {)\ x^{3}-2x^{2}+0x-4}}\\{\underline {x^{3}-3x^{2}{\color {White}{}+0x-4}}}\\+x^{2}+0x{\color {White}{}-4}\\{\underline {+x^{2}-3x{\color {White}{}-4}}}\\+3x-4\\\end{array}}}
534:
3676:{\displaystyle {\begin{array}{r}x-10\\x^{2}-2x+1\ {\overline {)\ x^{3}-12x^{2}+0x-42}}\\{\underline {x^{3}-{\color {White}0}2x^{2}+{\color {White}1}x}}{\color {White}{}-42}\\-10x^{2}-{\color {White}01}x-42\\{\underline {-10x^{2}+20x-10}}\\-21x-32\end{array}}}
1877:
2405:{\displaystyle {\begin{matrix}\qquad \qquad \quad {\bcancel {x^{2}}}\quad 3x\\\qquad \quad {\bcancel {x^{3}}}+{\bcancel {-2x^{2}}}+{\bcancel {0x}}-4\\{\underline {\div \qquad \qquad \qquad \qquad \qquad x-3}}\\x^{2}+x\qquad \end{matrix}}}
761:
Subtract the product just obtained from the appropriate terms of the original dividend (being careful that subtracting something having a minus sign is equivalent to adding something having a plus sign), and write the result underneath
1060:{\displaystyle {\begin{array}{l}{\color {White}x-3\ )\ x^{3}-2}x^{2}\\x-3\ {\overline {)\ x^{3}-2x^{2}+0x-4}}\\{\color {White}x-3\ )\ }{\underline {x^{3}-3x^{2}}}\\{\color {White}x-3\ )\ 0x^{3}}+{\color {White}}x^{2}+0x\end{array}}}
563:
3092:
This method is especially useful for cubic polynomials, and sometimes all the roots of a higher-degree polynomial can be obtained. For example, if the rational root theorem produces a single (rational) root of a
1911:
2193:{\displaystyle {\begin{matrix}\qquad x^{2}\\\qquad \quad {\bcancel {x^{3}}}+{\bcancel {-2x^{2}}}+{0x}-4\\{\underline {\div \qquad \qquad \qquad \qquad \qquad x-3}}\\x^{2}\qquad \qquad \end{matrix}}}
1905:
The division is at first written in a similar way as long multiplication with the dividend at the top, and the divisor below it. The quotient is to be written below the bar from left to right.
394:
3073:
3384:
3279:
360:
256:
3728:
1730:
2775:
292:
3308:
750:{\displaystyle {\begin{array}{l}{\color {White}x-3\ )\ x^{3}-2}x^{2}\\x-3\ {\overline {)\ x^{3}-2x^{2}+0x-4}}\\{\color {White}x-3\ )\ }x^{3}-3x^{2}\end{array}}}
1898:
Blomqvist's method is an abbreviated version of the long division above. This pen-and-paper method uses the same algorithm as polynomial long division, but
50:. It can be done easily by hand, because it separates an otherwise complex division problem into smaller ones. Sometimes using a shorthand version called
540:
Multiply the divisor by the result just obtained (the first term of the eventual quotient). Write the result under the first two terms of the dividend (
167:. Thus long division is a means for testing whether one polynomial has another as a factor, and, if it does, for factoring it out. For example, if a
3101:
can then be used to find the other four roots of the quintic. There is, however, no general way to solve a quintic by purely algebraic methods, see
3800:
2001:{\displaystyle {\begin{matrix}\qquad \qquad x^{3}-2x^{2}+{0x}-4\\{\underline {\div \quad \qquad \qquad \qquad \qquad x-3}}\end{matrix}}}
3776:
54:
is faster, with less writing and fewer calculations. Another abbreviated method is polynomial short division (Blomqvist's method).
3884:
3835:
529:{\displaystyle {\begin{array}{l}{\color {White}x-3\ )\ x^{3}-2}x^{2}\\x-3\ {\overline {)\ x^{3}-2x^{2}+0x-4}}\end{array}}}
4064:
2705:
58:
369:
Divide the first term of the dividend by the highest term of the divisor (meaning the one with the highest power of
2986:
4125:
1902:
is used to determine remainders. This requires less writing, and can therefore be a faster method once mastered.
1071:
Repeat the previous three steps, except this time use the two terms that have just been written as the dividend.
3097:, it can be factored out to obtain a quartic (fourth degree) quotient; the explicit formula for the roots of a
2634:
as follows, where +, −, and × represent polynomial arithmetic, and / represents simple division of two terms:
3316:
4059:
4043:
3750:
3223:
4079:
2027:
has been divided leaving no remainder, and can therefore be marked as used with a backslash. The result
307:
3924:
206:
4120:
3877:
2656:
t ← lead(r) / lead(d) // Divide the leading terms q ← q + t r ← r − t × d
3102:
164:
2690:; the region under the horizontal line is used to compute and write down the successive values of
4008:
3796:
3738:
3689:
1872:{\displaystyle {x^{3}-2x^{2}-4}=(x-3)\,\underbrace {(x^{2}+x+3)} _{q(x)}+\underbrace {5} _{r(x)}}
4038:
4033:
4028:
3906:
43:
4018:
3998:
2845:
2742:
3850:
Strickland-Constable, Charles, "A simple method for finding tangents to polynomial graphs",
4115:
4084:
4003:
3870:
3852:
2844:
Sometimes one or more roots of a polynomial are known, perhaps having been found using the
265:
168:
8:
3897:
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is written, term after term, above the horizontal line, the last term being the value of
4074:
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has been divided leaving no remainder, and can therefore be marked as used. The result
1899:
1886:
algorithm for arithmetic is very similar to the above algorithm, in which the variable
51:
20:
3919:
3831:
3761:
3741:
uses the remainder of polynomial division to detect errors in transmitted messages.
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2415:
Divide the highest term of the remainder by the highest term of the divisor (3x ÷
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2644:
require d ≠ 0 q ← 0 r ← n // At each step n = d × q + r
3953:
3948:
152:
3805:
3113:
Polynomial long division can be used to find the equation of the line that is
4109:
2203:
Divide the highest term of the remainder by the highest term of the divisor (
1883:
47:
4089:
3396:
2011:
Divide the first term of the dividend by the highest term of the divisor (
1383:
1081:
805:
568:
399:
3893:
2631:
39:
4023:
3220:
Find the equation of the line that is tangent to the following curve
2828:
35:
2678:
This algorithm describes exactly the above paper and pencil method:
46:, a generalized version of the familiar arithmetic technique called
4013:
3758:, a more concise method of performing Euclidean polynomial division
2912:) is simply the quotient obtained from the division process; since
83:
3114:
27:
3807:
Blomqvist's division: the simplest method for solving divisions?
365:
The quotient and remainder can then be determined as follows:
2864:
is known then polynomial long division can be used to factor
57:
Polynomial long division is an algorithm that implements the
2223:
is then multiplied by the second term in the divisor −3 = −3
2031:
is then multiplied by the second term in the divisor −3 = −3
1373:
Repeat step 4. This time, there is nothing to "bring down".
3108:
2800:) is the unique pair of polynomials having this property.
3892:
2803:
The process of getting the uniquely defined polynomials
203:
Find the quotient and the remainder of the division of
2430:
2258:
2066:
1916:
3692:
3394:
3319:
3290:
3226:
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2745:
2428:
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1381:
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1890:
is replaced (in base 10) by the specific number 10.
795:
Then, "bring down" the next term from the dividend.
2671:); in that case the result is just the trivial (0,
2035:. Determine the partial remainder by subtracting −2
3722:
3675:
3378:
3302:
3273:
3067:
2769:
2595:
2404:
2227:. Determine the partial remainder by subtracting 0
2192:
2000:
1871:
1695:
1361:
1059:
749:
528:
354:
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250:
144:do not depend on the method used to compute them.
2614:), and the number left over (5) is the remainder
2523:
2513:
2490:
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2453:
2438:
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2309:
2292:
2265:
2104:
2087:
1716:), and the number left over (5) is the remainder
4107:
2924:), it is known that the remainder must be zero.
3915:Zero polynomial (degree undefined or −1 or −∞)
3878:
2606:The polynomial below the bar is the quotient
1708:The polynomial above the bar is the quotient
178:is known, it can be factored out by dividing
3068:{\displaystyle (x-r)(x-s)=x^{2}-(r{+}s)x+rs}
2983:), etc. Alternatively, the quadratic factor
1893:
19:For a shorthand version of this method, see
301:The dividend is first rewritten like this:
198:
42:by another polynomial of the same or lower
3885:
3871:
3825:
3777:Greatest common divisor of two polynomials
3732:
3168:then the equation of the tangent line at
1786:
3109:Finding tangents to polynomial functions
2839:
2215:). Place the result (+x) below the bar.
2827:). Polynomial long division is thus an
1032:
4108:
3379:{\displaystyle (x-1)^{2}=(x^{2}-2x+1)}
3148:) is the remainder of the division of
61:, which starting from two polynomials
3866:
3591:
3556:
3540:
3517:
3313:Begin by dividing the polynomial by:
2697:
2243:as used and place the new remainder 3
1615:
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403:
16:Algorithm for division of polynomials
3274:{\displaystyle y=(x^{3}-12x^{2}-42)}
2722:≠ 0, polynomial division provides a
2663:This works equally well when degree(
2630:The algorithm can be represented in
2051:as used and place the new remainder
128:. These conditions uniquely define
2682:is written on the left of the ")";
2023:). Place the result below the bar.
377:). Place the result above the bar (
13:
355:{\displaystyle x^{3}-2x^{2}+0x-4.}
14:
4137:
3083:) to obtain a quotient of degree
2706:Euclidean division of polynomials
59:Euclidean division of polynomials
251:{\displaystyle (x^{3}-2x^{2}-4)}
2834:
2710:For every pair of polynomials (
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1:
4075:Horner's method of evaluation
3782:
3212:is a root of the polynomial.
3208:regardless of whether or not
3178:to the graph of the function
2955:can be divided out to obtain
2943:) are known, a linear factor
2625:
3751:Polynomial remainder theorem
3491:
3215:
2900:) is a polynomial of degree
1482:
1186:
923:
686:
517:
124:is lower than the degree of
7:
4080:Polynomial identity testing
3856:89, November 2005: 466-467.
3744:
3723:{\displaystyle y=(-21x-32)}
2927:Likewise, if several roots
10:
4142:
3830:. READ BOOKS. p. 24.
3121:defined by the polynomial
2703:
193:
18:
4052:
3991:
3904:
2916:is known to be a root of
1894:Polynomial short division
3129:) at a particular point
2831:for Euclidean division.
373:, which in this case is
199:Polynomial long division
32:polynomial long division
4065:Greatest common divisor
3739:cyclic redundancy check
3733:Cyclic redundancy check
2825:division transformation
2770:{\displaystyle A=BQ+R,}
4126:Division (mathematics)
3937:Quadratic function (2)
3724:
3677:
3380:
3304:
3275:
3075:can be divided out of
3069:
2975:can be divided out of
2771:
2652:degree(r) ≥ degree(d)
2597:
2406:
2194:
2002:
1873:
1697:
1363:
1061:
751:
530:
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288:
252:
3920:Constant function (0)
3725:
3686:The tangent line is
3678:
3381:
3305:
3276:
3119:graph of the function
3070:
2846:rational root theorem
2840:Factoring polynomials
2772:
2598:
2407:
2195:
2003:
1874:
1698:
1364:
1062:
752:
531:
357:
289:
287:{\displaystyle (x-3)}
253:
120:= 0 or the degree of
4053:Tools and algorithms
3973:Quintic function (5)
3961:Quartic function (4)
3898:polynomial functions
3853:Mathematical Gazette
3690:
3392:
3317:
3288:
3224:
3103:Abel–Ruffini theorem
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3983:Septic equation (7)
3978:Sextic equation (6)
3925:Linear function (1)
3826:S. Barnard (2008).
3303:{\displaystyle x=1}
136:, which means that
3949:Cubic function (3)
3942:Quadratic equation
3756:Synthetic division
3720:
3673:
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3553:
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3099:quartic polynomial
3095:quintic polynomial
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2821:Euclidean division
2767:
2698:Euclidean division
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1998:
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1900:mental calculation
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52:synthetic division
21:synthetic division
4103:
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4044:Quasi-homogeneous
3837:978-1-4437-3086-0
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4121:Computer algebra
3966:Quartic equation
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3930:Linear equation
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3801:Wayback Machine
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2873:
2842:
2837:
2744:
2741:
2740:
2708:
2700:
2693:
2689:
2685:
2681:
2661:
2628:
2590:
2589:
2570:
2566:
2563:
2562:
2538:
2536:
2533:
2532:
2522:
2512:
2501:
2497:
2489:
2478:
2474:
2472:
2467:
2466:
2452:
2443:
2439:
2437:
2429:
2427:
2424:
2423:
2399:
2398:
2385:
2381:
2378:
2377:
2353:
2351:
2348:
2347:
2331:
2320:
2316:
2308:
2297:
2293:
2291:
2286:
2285:
2270:
2266:
2264:
2257:
2255:
2252:
2251:
2187:
2186:
2178:
2174:
2171:
2170:
2146:
2144:
2141:
2140:
2126:
2115:
2111:
2103:
2092:
2088:
2086:
2081:
2080:
2074:
2070:
2065:
2063:
2060:
2059:
1995:
1994:
1970:
1968:
1965:
1964:
1950:
1941:
1937:
1925:
1921:
1915:
1913:
1910:
1909:
1896:
1854:
1844:
1825:
1798:
1794:
1790:
1788:
1755:
1751:
1739:
1735:
1734:
1732:
1729:
1728:
1690:
1689:
1680:
1679:
1657:
1655:
1652:
1651:
1633:
1632:
1616:
1614:
1599:
1595:
1591:
1589:
1586:
1585:
1575:
1573:
1558:
1554:
1548:
1547:
1522:
1520:
1514:
1510:
1498:
1494:
1493:
1491:
1488:
1487:
1460:
1456:
1444:
1440:
1433:
1431:
1416:
1415:
1399:
1390:
1386:
1382:
1380:
1377:
1376:
1356:
1355:
1337:
1336:
1320:
1318:
1303:
1299:
1295:
1293:
1290:
1289:
1279:
1277:
1262:
1258:
1252:
1251:
1226:
1224:
1218:
1214:
1202:
1198:
1197:
1195:
1192:
1191:
1164:
1160:
1148:
1144:
1137:
1135:
1120:
1119:
1109:
1107:
1097:
1088:
1084:
1080:
1078:
1075:
1074:
1054:
1053:
1038:
1034:
1031:
1020:
1016:
993:
990:
989:
977:
973:
961:
957:
956:
954:
932:
929:
928:
901:
897:
885:
881:
874:
872:
857:
856:
850:
846:
832:
828:
808:
804:
802:
799:
798:
765:
763:
744:
743:
737:
733:
721:
717:
695:
692:
691:
664:
660:
648:
644:
637:
635:
620:
619:
613:
609:
595:
591:
571:
567:
565:
562:
561:
541:
523:
522:
495:
491:
479:
475:
468:
466:
451:
450:
444:
440:
426:
422:
402:
398:
396:
393:
392:
331:
327:
315:
311:
309:
306:
305:
267:
264:
263:
233:
229:
217:
213:
208:
205:
204:
201:
196:
155:the polynomial
81:is not zero, a
77:) produces, if
38:for dividing a
24:
17:
12:
11:
5:
4139:
4129:
4128:
4123:
4118:
4101:
4100:
4098:
4097:
4092:
4087:
4082:
4077:
4072:
4067:
4062:
4056:
4054:
4050:
4049:
4047:
4046:
4041:
4036:
4031:
4026:
4021:
4016:
4011:
4006:
4001:
3995:
3993:
3989:
3988:
3986:
3985:
3980:
3975:
3970:
3969:
3968:
3958:
3957:
3956:
3954:Cubic equation
3946:
3945:
3944:
3934:
3933:
3932:
3922:
3917:
3911:
3909:
3902:
3901:
3890:
3889:
3882:
3875:
3867:
3859:
3858:
3843:
3836:
3828:Higher Algebra
3818:
3787:
3786:
3784:
3781:
3780:
3779:
3774:
3769:
3764:
3762:Ruffini's rule
3759:
3753:
3746:
3743:
3734:
3731:
3719:
3716:
3713:
3710:
3707:
3704:
3701:
3698:
3695:
3684:
3683:
3668:
3665:
3662:
3659:
3656:
3653:
3652:
3647:
3643:
3640:
3637:
3634:
3631:
3626:
3622:
3618:
3615:
3609:
3608:
3605:
3602:
3599:
3594:
3589:
3584:
3580:
3576:
3573:
3570:
3569:
3564:
3561:
3552:
3548:
3543:
3538:
3533:
3529:
3525:
3520:
3515:
3510:
3506:
3499:
3498:
3493:
3489:
3486:
3483:
3480:
3477:
3472:
3468:
3464:
3461:
3456:
3452:
3445:
3436:
3433:
3430:
3427:
3424:
3419:
3415:
3411:
3410:
3407:
3404:
3401:
3398:
3397:
3375:
3372:
3369:
3366:
3363:
3360:
3355:
3351:
3347:
3344:
3339:
3335:
3331:
3328:
3325:
3322:
3311:
3310:
3299:
3296:
3293:
3270:
3267:
3264:
3259:
3255:
3251:
3248:
3243:
3239:
3235:
3232:
3229:
3217:
3214:
3110:
3107:
3064:
3061:
3058:
3055:
3052:
3049:
3045:
3041:
3038:
3035:
3030:
3026:
3022:
3019:
3016:
3013:
3010:
3007:
3004:
3001:
2998:
2995:
2992:
2848:. If one root
2841:
2838:
2836:
2833:
2788:) < degree(
2778:
2777:
2766:
2763:
2760:
2757:
2754:
2751:
2748:
2704:Main article:
2699:
2696:
2691:
2687:
2683:
2679:
2667:) < degree(
2636:
2627:
2624:
2604:
2603:
2587:
2584:
2581:
2578:
2573:
2569:
2565:
2564:
2559:
2555:
2552:
2549:
2541:
2535:
2534:
2529:
2526:
2519:
2516:
2511:
2504:
2500:
2496:
2493:
2488:
2481:
2477:
2469:
2468:
2465:
2459:
2456:
2446:
2442:
2432:
2431:
2413:
2412:
2396:
2393:
2388:
2384:
2380:
2379:
2374:
2370:
2367:
2364:
2356:
2350:
2349:
2346:
2343:
2338:
2335:
2330:
2323:
2319:
2315:
2312:
2307:
2300:
2296:
2288:
2287:
2284:
2281:
2273:
2269:
2260:
2259:
2201:
2200:
2181:
2177:
2173:
2172:
2167:
2163:
2160:
2157:
2149:
2143:
2142:
2139:
2136:
2132:
2129:
2125:
2118:
2114:
2110:
2107:
2102:
2095:
2091:
2083:
2082:
2077:
2073:
2068:
2067:
2009:
2008:
1991:
1987:
1984:
1981:
1973:
1967:
1966:
1963:
1960:
1956:
1953:
1949:
1944:
1940:
1936:
1933:
1928:
1924:
1918:
1917:
1895:
1892:
1880:
1879:
1866:
1863:
1860:
1857:
1851:
1848:
1842:
1837:
1834:
1831:
1828:
1822:
1818:
1815:
1812:
1809:
1806:
1801:
1797:
1793:
1785:
1782:
1779:
1776:
1773:
1770:
1766:
1763:
1758:
1754:
1750:
1747:
1742:
1738:
1706:
1705:
1704:
1703:
1688:
1685:
1682:
1681:
1676:
1672:
1669:
1666:
1663:
1660:
1654:
1653:
1650:
1647:
1644:
1641:
1638:
1635:
1634:
1629:
1623:
1620:
1613:
1610:
1607:
1602:
1598:
1594:
1588:
1587:
1582:
1579:
1572:
1569:
1566:
1561:
1557:
1553:
1550:
1549:
1544:
1538:
1535:
1532:
1529:
1526:
1517:
1513:
1509:
1506:
1501:
1497:
1490:
1489:
1484:
1480:
1477:
1474:
1471:
1468:
1463:
1459:
1455:
1452:
1447:
1443:
1436:
1427:
1424:
1421:
1418:
1417:
1414:
1411:
1408:
1403:
1398:
1393:
1389:
1385:
1384:
1371:
1370:
1369:
1354:
1351:
1348:
1345:
1342:
1339:
1338:
1333:
1327:
1324:
1317:
1314:
1311:
1306:
1302:
1298:
1292:
1291:
1286:
1283:
1276:
1273:
1270:
1265:
1261:
1257:
1254:
1253:
1248:
1242:
1239:
1236:
1233:
1230:
1221:
1217:
1213:
1210:
1205:
1201:
1194:
1193:
1188:
1184:
1181:
1178:
1175:
1172:
1167:
1163:
1159:
1156:
1151:
1147:
1140:
1131:
1128:
1125:
1122:
1121:
1116:
1113:
1106:
1101:
1096:
1091:
1087:
1083:
1082:
1069:
1068:
1067:
1052:
1049:
1046:
1041:
1037:
1030:
1023:
1019:
1015:
1009:
1003:
1000:
997:
992:
991:
986:
980:
976:
972:
969:
964:
960:
948:
942:
939:
936:
931:
930:
925:
921:
918:
915:
912:
909:
904:
900:
896:
893:
888:
884:
877:
868:
865:
862:
859:
858:
853:
849:
843:
840:
835:
831:
824:
818:
815:
812:
807:
806:
759:
758:
757:
740:
736:
732:
729:
724:
720:
711:
705:
702:
699:
694:
693:
688:
684:
681:
678:
675:
672:
667:
663:
659:
656:
651:
647:
640:
631:
628:
625:
622:
621:
616:
612:
606:
603:
598:
594:
587:
581:
578:
575:
570:
569:
538:
537:
536:
519:
515:
512:
509:
506:
503:
498:
494:
490:
487:
482:
478:
471:
462:
459:
456:
453:
452:
447:
443:
437:
434:
429:
425:
418:
412:
409:
406:
401:
400:
363:
362:
351:
348:
345:
342:
339:
334:
330:
326:
323:
318:
314:
283:
280:
277:
274:
271:
247:
244:
241:
236:
232:
228:
225:
220:
216:
212:
200:
197:
195:
192:
153:if and only if
114:
113:
15:
9:
6:
4:
3:
2:
4138:
4127:
4124:
4122:
4119:
4117:
4114:
4113:
4111:
4096:
4095:Gröbner basis
4093:
4091:
4088:
4086:
4083:
4081:
4078:
4076:
4073:
4071:
4068:
4066:
4063:
4061:
4060:Factorization
4058:
4057:
4055:
4051:
4045:
4042:
4040:
4037:
4035:
4032:
4030:
4027:
4025:
4022:
4020:
4017:
4015:
4012:
4010:
4007:
4005:
4002:
4000:
3997:
3996:
3994:
3992:By properties
3990:
3984:
3981:
3979:
3976:
3974:
3971:
3967:
3964:
3963:
3962:
3959:
3955:
3952:
3951:
3950:
3947:
3943:
3940:
3939:
3938:
3935:
3931:
3928:
3927:
3926:
3923:
3921:
3918:
3916:
3913:
3912:
3910:
3908:
3903:
3899:
3895:
3888:
3883:
3881:
3876:
3874:
3869:
3868:
3865:
3855:
3854:
3847:
3839:
3833:
3829:
3822:
3809:
3808:
3802:
3798:
3792:
3788:
3778:
3775:
3773:
3772:Gröbner basis
3770:
3768:
3765:
3763:
3760:
3757:
3754:
3752:
3749:
3748:
3742:
3740:
3730:
3714:
3711:
3708:
3705:
3702:
3696:
3693:
3666:
3663:
3660:
3657:
3654:
3645:
3641:
3638:
3635:
3632:
3629:
3624:
3620:
3616:
3613:
3603:
3600:
3597:
3592:
3587:
3582:
3578:
3574:
3571:
3562:
3559:
3550:
3546:
3541:
3536:
3531:
3527:
3523:
3518:
3513:
3508:
3504:
3487:
3484:
3481:
3478:
3475:
3470:
3466:
3462:
3459:
3454:
3450:
3434:
3431:
3428:
3425:
3422:
3417:
3413:
3405:
3402:
3399:
3388:
3387:
3386:
3370:
3367:
3364:
3361:
3358:
3353:
3349:
3342:
3337:
3329:
3326:
3323:
3297:
3294:
3291:
3283:
3282:
3281:
3265:
3262:
3257:
3253:
3249:
3246:
3241:
3237:
3230:
3227:
3213:
3211:
3205:
3201:
3197:
3190:
3186:
3182:
3176:
3172:
3165:
3161:
3155:
3151:
3147:
3143:
3137:
3133:
3128:
3124:
3120:
3116:
3106:
3104:
3100:
3096:
3090:
3087:
3082:
3078:
3062:
3059:
3056:
3053:
3047:
3043:
3039:
3033:
3028:
3024:
3020:
3014:
3011:
3008:
2999:
2996:
2993:
2982:
2978:
2972:
2968:
2962:
2958:
2952:
2948:
2942:
2938:
2934:
2930:
2925:
2923:
2919:
2915:
2911:
2907:
2903:
2899:
2895:
2889:
2885:
2881:
2877:
2871:
2867:
2863:
2859:
2855:
2851:
2847:
2832:
2830:
2826:
2822:
2818:
2814:
2810:
2806:
2801:
2799:
2795:
2792:). Moreover (
2791:
2787:
2784:=0 or degree(
2783:
2764:
2761:
2758:
2755:
2752:
2749:
2746:
2739:
2738:
2737:
2735:
2732:
2728:
2725:
2721:
2717:
2713:
2707:
2702:
2695:
2676:
2674:
2670:
2666:
2659:
2655:
2651:
2647:
2643:
2639:
2635:
2633:
2623:
2621:
2617:
2613:
2609:
2585:
2582:
2579:
2576:
2571:
2567:
2557:
2553:
2550:
2547:
2539:
2527:
2524:
2517:
2514:
2509:
2502:
2498:
2494:
2491:
2486:
2479:
2475:
2463:
2457:
2454:
2444:
2440:
2422:
2421:
2420:
2418:
2394:
2391:
2386:
2382:
2372:
2368:
2365:
2362:
2354:
2344:
2341:
2336:
2333:
2328:
2321:
2317:
2313:
2310:
2305:
2298:
2294:
2282:
2279:
2271:
2267:
2250:
2249:
2248:
2246:
2242:
2238:
2234:
2230:
2226:
2222:
2218:
2214:
2210:
2206:
2179:
2175:
2165:
2161:
2158:
2155:
2147:
2137:
2134:
2130:
2127:
2123:
2116:
2112:
2108:
2105:
2100:
2093:
2089:
2075:
2071:
2058:
2057:
2056:
2054:
2050:
2046:
2042:
2038:
2034:
2030:
2026:
2022:
2018:
2014:
1989:
1985:
1982:
1979:
1971:
1961:
1958:
1954:
1951:
1947:
1942:
1938:
1934:
1931:
1926:
1922:
1908:
1907:
1906:
1903:
1901:
1891:
1889:
1885:
1884:long division
1861:
1855:
1849:
1846:
1840:
1832:
1826:
1820:
1813:
1810:
1807:
1804:
1799:
1795:
1780:
1777:
1774:
1768:
1764:
1761:
1756:
1752:
1748:
1745:
1740:
1736:
1727:
1726:
1725:
1723:
1719:
1715:
1711:
1686:
1683:
1674:
1670:
1667:
1664:
1661:
1658:
1648:
1645:
1642:
1639:
1636:
1627:
1621:
1618:
1611:
1608:
1605:
1600:
1596:
1592:
1580:
1577:
1570:
1567:
1564:
1559:
1555:
1551:
1542:
1536:
1533:
1530:
1527:
1524:
1515:
1511:
1507:
1504:
1499:
1495:
1478:
1475:
1472:
1469:
1466:
1461:
1457:
1453:
1450:
1445:
1441:
1425:
1422:
1419:
1412:
1409:
1406:
1401:
1396:
1391:
1387:
1375:
1374:
1372:
1352:
1349:
1346:
1343:
1340:
1331:
1325:
1322:
1315:
1312:
1309:
1304:
1300:
1296:
1284:
1281:
1274:
1271:
1268:
1263:
1259:
1255:
1246:
1240:
1237:
1234:
1231:
1228:
1219:
1215:
1211:
1208:
1203:
1199:
1182:
1179:
1176:
1173:
1170:
1165:
1161:
1157:
1154:
1149:
1145:
1129:
1126:
1123:
1114:
1111:
1104:
1099:
1094:
1089:
1085:
1073:
1072:
1070:
1050:
1047:
1044:
1039:
1035:
1028:
1021:
1017:
1013:
1001:
998:
995:
984:
978:
974:
970:
967:
962:
958:
940:
937:
934:
919:
916:
913:
910:
907:
902:
898:
894:
891:
886:
882:
866:
863:
860:
851:
847:
841:
838:
833:
829:
816:
813:
810:
797:
796:
792:
788:
784:
780:
776:
772:
768:
760:
738:
734:
730:
727:
722:
718:
703:
700:
697:
682:
679:
676:
673:
670:
665:
661:
657:
654:
649:
645:
629:
626:
623:
614:
610:
604:
601:
596:
592:
579:
576:
573:
560:
559:
556:
552:
548:
544:
539:
513:
510:
507:
504:
501:
496:
492:
488:
485:
480:
476:
460:
457:
454:
445:
441:
435:
432:
427:
423:
410:
407:
404:
391:
390:
388:
384:
380:
376:
372:
368:
367:
366:
349:
346:
343:
340:
337:
332:
328:
324:
321:
316:
312:
304:
303:
302:
299:
297:
278:
275:
272:
261:
242:
239:
234:
230:
226:
223:
218:
214:
191:
189:
186: –
185:
181:
177:
173:
170:
166:
162:
158:
154:
150:
145:
143:
139:
135:
131:
127:
123:
119:
111:
107:
103:
100:
99:
98:
96:
93:
89:
86:
85:
80:
76:
72:
68:
64:
60:
55:
53:
49:
48:long division
45:
41:
37:
33:
29:
22:
4090:Discriminant
4069:
4009:Multivariate
3851:
3846:
3827:
3821:
3811:, retrieved
3806:
3797:Ghostarchive
3795:Archived at
3791:
3736:
3685:
3312:
3219:
3209:
3203:
3199:
3195:
3188:
3184:
3180:
3174:
3170:
3163:
3159:
3153:
3149:
3145:
3141:
3135:
3131:
3126:
3122:
3112:
3091:
3085:
3080:
3076:
2980:
2976:
2970:
2966:
2963:), and then
2960:
2956:
2950:
2946:
2940:
2936:
2935:, . . . of
2932:
2928:
2926:
2921:
2917:
2913:
2909:
2905:
2901:
2897:
2893:
2887:
2883:
2879:
2875:
2869:
2865:
2861:
2860:) of degree
2857:
2853:
2849:
2843:
2835:Applications
2824:
2820:
2816:
2812:
2808:
2804:
2802:
2797:
2793:
2789:
2785:
2781:
2779:
2733:
2730:
2726:
2723:
2719:
2718:) such that
2715:
2711:
2709:
2701:
2677:
2672:
2668:
2664:
2662:
2657:
2653:
2649:
2645:
2641:
2637:
2629:
2619:
2615:
2611:
2607:
2605:
2416:
2414:
2244:
2240:
2236:
2232:
2228:
2224:
2220:
2216:
2212:
2208:
2204:
2202:
2052:
2048:
2044:
2040:
2036:
2032:
2028:
2024:
2020:
2016:
2012:
2010:
1904:
1897:
1887:
1881:
1721:
1717:
1713:
1709:
1707:
790:
786:
782:
778:
774:
770:
766:
554:
550:
546:
542:
386:
382:
378:
374:
370:
364:
300:
295:
259:
202:
187:
183:
179:
175:
171:
160:
156:
148:
146:
141:
137:
133:
129:
125:
121:
117:
115:
109:
105:
101:
94:
91:
87:
82:
78:
74:
70:
66:
62:
56:
31:
25:
4116:Polynomials
4039:Homogeneous
4034:Square-free
4029:Irreducible
3894:Polynomials
2823:(sometimes
2780:and either
2736:such that
151:= 0 occurs
147:The result
116:and either
4110:Categories
3999:Univariate
3813:2019-12-10
3783:References
2819:is called
2632:pseudocode
2626:Pseudocode
2247:above it.
2055:above it.
97:such that
40:polynomial
4085:Resultant
4024:Trinomial
4004:Bivariate
3712:−
3703:−
3664:−
3655:−
3646:_
3639:−
3614:−
3601:−
3588:−
3572:−
3560:−
3551:_
3514:−
3492:¯
3485:−
3460:−
3423:−
3403:−
3359:−
3327:−
3263:−
3247:−
3034:−
3012:−
2997:−
2829:algorithm
2731:remainder
2558:_
2551:−
2540:÷
2525:−
2492:−
2373:_
2366:−
2355:÷
2342:−
2311:−
2166:_
2159:−
2148:÷
2135:−
2106:−
2047:. Mark −2
1990:_
1983:−
1972:÷
1959:−
1932:−
1850:⏟
1821:⏟
1778:−
1762:−
1746:−
1675:_
1668:−
1646:−
1628:_
1619:−
1606:−
1578:−
1543:_
1534:−
1505:−
1483:¯
1476:−
1451:−
1423:−
1350:−
1332:_
1323:−
1310:−
1282:−
1247:_
1238:−
1209:−
1187:¯
1180:−
1155:−
1127:−
999:−
985:_
968:−
938:−
924:¯
917:−
892:−
864:−
839:−
814:−
728:−
701:−
687:¯
680:−
655:−
627:−
602:−
577:−
518:¯
511:−
486:−
458:−
433:−
408:−
347:−
322:−
276:−
240:−
224:−
92:remainder
36:algorithm
4070:Division
4019:Binomial
4014:Monomial
3799:and the
3745:See also
2724:quotient
2638:function
2239:. Mark 0
260:dividend
84:quotient
67:dividend
3216:Example
3117:to the
3115:tangent
2660:(q, r)
549:− 3) =
296:divisor
194:Example
75:divisor
28:algebra
3907:degree
3834:
3447:
3438:
2904:− 1.
2892:where
2729:and a
2658:return
2648:r ≠ 0
2640:n / d
1438:
1429:
1142:
1133:
1011:
1005:
950:
944:
879:
870:
826:
820:
781:) = −2
713:
707:
642:
633:
589:
583:
473:
464:
420:
414:
294:, the
258:, the
165:factor
90:and a
69:) and
44:degree
34:is an
3156:) by
2811:from
2646:while
2235:) = 3
2231:− (−3
2039:− (−3
773:) − (
262:, by
163:as a
73:(the
65:(the
3896:and
3832:ISBN
3284:at:
3088:− 2.
2815:and
2807:and
2043:) =
1882:The
182:by (
169:root
159:has
140:and
132:and
3905:By
3193:is
3140:If
2675:).
2650:and
2622:).
1724:).
785:+ 3
777:− 3
769:− 2
558:).
553:− 3
545:· (
389:).
190:).
174:of
26:In
4112::
3803::
3737:A
3715:32
3706:21
3667:32
3658:21
3642:10
3633:20
3617:10
3604:42
3593:01
3575:10
3563:42
3488:42
3463:12
3406:10
3266:42
3250:12
3206:),
3198:=
3183:=
3173:=
3166:),
3162:–
3134:=
3105:.
2969:−
2949:−
2931:,
2878:−
2796:,
2714:,
2694:.
2654:do
2642:is
2211:=
2207:÷
2019:=
2015:÷
789:=
385:=
381:÷
350:4.
298:.
108:+
106:BQ
104:=
30:,
3886:e
3879:t
3872:v
3840:.
3718:)
3709:x
3700:(
3697:=
3694:y
3661:x
3636:x
3630:+
3625:2
3621:x
3598:x
3583:2
3579:x
3547:x
3542:1
3537:+
3532:2
3528:x
3524:2
3519:0
3509:3
3505:x
3482:x
3479:0
3476:+
3471:2
3467:x
3455:3
3451:x
3444:)
3435:1
3432:+
3429:x
3426:2
3418:2
3414:x
3400:x
3374:)
3371:1
3368:+
3365:x
3362:2
3354:2
3350:x
3346:(
3343:=
3338:2
3334:)
3330:1
3324:x
3321:(
3298:1
3295:=
3292:x
3269:)
3258:2
3254:x
3242:3
3238:x
3234:(
3231:=
3228:y
3210:r
3204:x
3202:(
3200:R
3196:y
3191:)
3189:x
3187:(
3185:P
3181:y
3175:r
3171:x
3164:r
3160:x
3158:(
3154:x
3152:(
3150:P
3146:x
3144:(
3142:R
3138:.
3136:r
3132:x
3127:x
3125:(
3123:P
3086:n
3081:x
3079:(
3077:P
3063:s
3060:r
3057:+
3054:x
3051:)
3048:s
3044:+
3040:r
3037:(
3029:2
3025:x
3021:=
3018:)
3015:s
3009:x
3006:(
3003:)
3000:r
2994:x
2991:(
2981:x
2979:(
2977:Q
2973:)
2971:s
2967:x
2965:(
2961:x
2959:(
2957:Q
2953:)
2951:r
2947:x
2945:(
2941:x
2939:(
2937:P
2933:s
2929:r
2922:x
2920:(
2918:P
2914:r
2910:x
2908:(
2906:Q
2902:n
2898:x
2896:(
2894:Q
2890:)
2888:x
2886:(
2884:Q
2882:)
2880:r
2876:x
2874:(
2870:x
2868:(
2866:P
2862:n
2858:x
2856:(
2854:P
2850:r
2817:B
2813:A
2809:R
2805:Q
2798:R
2794:Q
2790:B
2786:R
2782:R
2765:,
2762:R
2759:+
2756:Q
2753:B
2750:=
2747:A
2734:R
2727:Q
2720:B
2716:B
2712:A
2692:r
2688:t
2684:q
2680:d
2673:n
2669:d
2665:n
2620:x
2618:(
2616:r
2612:x
2610:(
2608:q
2586:3
2583:+
2580:x
2577:+
2572:2
2568:x
2554:3
2548:x
2528:4
2518:x
2515:0
2510:+
2503:2
2499:x
2495:2
2487:+
2480:3
2476:x
2464:5
2458:x
2455:3
2445:2
2441:x
2417:x
2395:x
2392:+
2387:2
2383:x
2369:3
2363:x
2345:4
2337:x
2334:0
2329:+
2322:2
2318:x
2314:2
2306:+
2299:3
2295:x
2283:x
2280:3
2272:2
2268:x
2245:x
2241:x
2237:x
2233:x
2229:x
2225:x
2221:x
2217:x
2213:x
2209:x
2205:x
2180:2
2176:x
2162:3
2156:x
2138:4
2131:x
2128:0
2124:+
2117:2
2113:x
2109:2
2101:+
2094:3
2090:x
2076:2
2072:x
2053:x
2049:x
2045:x
2041:x
2037:x
2033:x
2029:x
2025:x
2021:x
2017:x
2013:x
1986:3
1980:x
1962:4
1955:x
1952:0
1948:+
1943:2
1939:x
1935:2
1927:3
1923:x
1888:x
1865:)
1862:x
1859:(
1856:r
1847:5
1841:+
1836:)
1833:x
1830:(
1827:q
1817:)
1814:3
1811:+
1808:x
1805:+
1800:2
1796:x
1792:(
1784:)
1781:3
1775:x
1772:(
1769:=
1765:4
1757:2
1753:x
1749:2
1741:3
1737:x
1722:x
1720:(
1718:r
1714:x
1712:(
1710:q
1687:5
1684:+
1671:9
1665:x
1662:3
1659:+
1649:4
1643:x
1640:3
1637:+
1622:4
1612:x
1609:3
1601:2
1597:x
1593:+
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1568:0
1565:+
1560:2
1556:x
1552:+
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1531:x
1528:0
1525:+
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1512:x
1508:3
1500:3
1496:x
1479:4
1473:x
1470:0
1467:+
1462:2
1458:x
1454:2
1446:3
1442:x
1435:)
1426:3
1420:x
1413:3
1410:+
1407:x
1402:1
1397:+
1392:2
1388:x
1353:4
1347:x
1344:3
1341:+
1326:4
1316:x
1313:3
1305:2
1301:x
1297:+
1285:4
1275:x
1272:0
1269:+
1264:2
1260:x
1256:+
1241:4
1235:x
1232:0
1229:+
1220:2
1216:x
1212:3
1204:3
1200:x
1183:4
1177:x
1174:0
1171:+
1166:2
1162:x
1158:2
1150:3
1146:x
1139:)
1130:3
1124:x
1115:3
1112:+
1105:x
1100:1
1095:+
1090:2
1086:x
1051:x
1048:0
1045:+
1040:2
1036:x
1029:+
1022:3
1018:x
1014:0
1008:)
1002:3
996:x
979:2
975:x
971:3
963:3
959:x
947:)
941:3
935:x
920:4
914:x
911:0
908:+
903:2
899:x
895:2
887:3
883:x
876:)
867:3
861:x
852:2
848:x
842:2
834:3
830:x
823:)
817:3
811:x
791:x
787:x
783:x
779:x
775:x
771:x
767:x
764:(
739:2
735:x
731:3
723:3
719:x
710:)
704:3
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683:4
677:x
674:0
671:+
666:2
662:x
658:2
650:3
646:x
639:)
630:3
624:x
615:2
611:x
605:2
597:3
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586:)
580:3
574:x
555:x
551:x
547:x
543:x
514:4
508:x
505:0
502:+
497:2
493:x
489:2
481:3
477:x
470:)
461:3
455:x
446:2
442:x
436:2
428:3
424:x
417:)
411:3
405:x
387:x
383:x
379:x
375:x
371:x
344:x
341:0
338:+
333:2
329:x
325:2
317:3
313:x
282:)
279:3
273:x
270:(
246:)
243:4
235:2
231:x
227:2
219:3
215:x
211:(
188:r
184:x
180:A
176:A
172:r
161:B
157:A
149:R
142:R
138:Q
134:R
130:Q
126:B
122:R
118:R
112:,
110:R
102:A
95:R
88:Q
79:B
71:B
63:A
23:.
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