2595:
1695:
1361:
3594:
2404:
1059:
2419:
1372:
2192:
1070:
3304:
2413:= 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.
2247:
749:
794:
2590:{\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}}}
2000:
1690:{\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}}}
2055:
1356:{\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}}}
528:
3589:{\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}}}
1871:
2399:{\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}}}
755:
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
1054:{\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}}}
557:
3086:
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
1905:
2187:{\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}}}
1899:
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.
388:
3067:
3277:
354:
253:
1724:
2769:
286:
744:{\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}}}
1892:
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
3657:
534:
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
3095:
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
3765:
3716:
3681:
1995:{\displaystyle {\begin{matrix}\qquad \qquad x^{3}-2x^{2}+{0x}-4\\{\underline {\div \quad \qquad \qquad \qquad \qquad x-3}}\end{matrix}}}
3945:
2699:
58:
54:
is faster, with less writing and fewer calculations. Another abbreviated method is polynomial short division (Blomqvist's method).
4006:
523:{\displaystyle {\begin{array}{l}{\color {White}x-3\ )\ x^{3}-2}x^{2}\\x-3\ {\overline {)\ x^{3}-2x^{2}+0x-4}}\end{array}}}
3940:
3924:
3631:
3960:
363:
Divide the first term of the dividend by the highest term of the divisor (meaning the one with the highest power of
3805:
2980:
4001:
3758:
1896:
is used to determine remainders. This requires less writing, and can therefore be a faster method once mastered.
1065:
Repeat the previous three steps, except this time use the two terms that have just been written as the dividend.
3096:
3091:, it can be factored out to obtain a quartic (fourth degree) quotient; the explicit formula for the roots of a
164:
2628:
as follows, where +, −, and × represent polynomial arithmetic, and / represents simple division of two terms:
3889:
3619:
3919:
3914:
3909:
3787:
43:
3227:
3899:
3879:
2839:
2021:
has been divided leaving no remainder, and can therefore be marked as used with a backslash. The result
301:
3996:
3965:
3884:
3751:
3733:
206:
168:
2650:
t ← lead(r) / lead(d) // Divide the leading terms q ← q + t r ← r − t × d
3778:
3112:
2684:; the region under the horizontal line is used to compute and write down the successive values of
3822:
3817:
3677:
3636:
1893:
1866:{\displaystyle {x^{3}-2x^{2}-4}=(x-3)\,\underbrace {(x^{2}+x+3)} _{q(x)}+\underbrace {5} _{r(x)}}
51:
20:
3955:
3800:
3712:
3642:
2736:
3975:
3853:
3846:
3841:
3731:
Strickland-Constable, Charles, "A simple method for finding tangents to polynomial graphs",
3652:
3647:
3092:
3088:
2838:
Sometimes one or more roots of a polynomial are known, perhaps having been found using the
3863:
3858:
3810:
3795:
262:
2680:
is written, term after term, above the horizontal line, the last term being the value of
3834:
3829:
3743:
2213:
has been divided leaving no remainder, and can therefore be marked as used. The result
1880:
algorithm for arithmetic is very similar to the above algorithm, in which the variable
152:
3990:
1877:
47:
3622:
uses the remainder of polynomial division to detect errors in transmitted messages.
3970:
2409:
Divide the highest term of the remainder by the highest term of the divisor (3x ÷
2638:
require d ≠ 0 q ← 0 r ← n // At each step n = d × q + r
3774:
3686:
3107:
Polynomial long division can be used to find the equation of the line that is
2625:
39:
2197:
Divide the highest term of the remainder by the highest term of the divisor (
3904:
2822:
35:
3309:
2005:
Divide the first term of the dividend by the highest term of the divisor (
1377:
1075:
3894:
799:
562:
393:
83:
3214:
Find the equation of the line that is tangent to the following curve at
3108:
27:
2672:
This algorithm describes exactly the above paper and pencil method:
46:, a generalized version of the familiar arithmetic technique called
3639:, a more concise method of performing Euclidean polynomial division
2906:) is simply the quotient obtained from the division process; since
3688:
Blomqvist's division: the simplest method for solving divisions?
3747:
359:
The quotient and remainder can then be determined as follows:
2858:
is known then polynomial long division can be used to factor
57:
Polynomial long division is an algorithm that implements the
2217:
is then multiplied by the second term in the divisor −3 = −3
2025:
is then multiplied by the second term in the divisor −3 = −3
1367:
Repeat step 4. This time, there is nothing to "bring down".
2794:) is the unique pair of polynomials having this property.
2797:
The process of getting the uniquely defined polynomials
203:
Find the quotient and the remainder of the division of
789:). Then, "bring down" the next term from the dividend.
2424:
2252:
2060:
1910:
3307:
3230:
2983:
2739:
2422:
2250:
2058:
1908:
1727:
1375:
1073:
797:
560:
391:
304:
265:
209:
1884:
is replaced (in base 10) by the specific number 10.
3933:
3872:
3785:
2665:); in that case the result is just the trivial (0,
2029:. Determine the partial remainder by subtracting −2
3588:
3271:
3061:
2763:
2589:
2398:
2221:. Determine the partial remainder by subtracting 0
2186:
1994:
1865:
1689:
1355:
1053:
743:
522:
348:
280:
247:
144:do not depend on the method used to compute them.
2608:), and the number left over (5) is the remainder
2517:
2507:
2484:
2467:
2447:
2432:
2326:
2303:
2286:
2259:
2098:
2081:
1710:), and the number left over (5) is the remainder
2918:), it is known that the remainder must be zero.
3796:Zero polynomial (degree undefined or −1 or −∞)
3759:
2600:The polynomial below the bar is the quotient
1702:The polynomial above the bar is the quotient
178:is known, it can be factored out by dividing
8:
3062:{\displaystyle (x-r)(x-s)=x^{2}-(r{+}s)x+rs}
2977:), etc. Alternatively, the quadratic factor
19:For a shorthand version of this method, see
295:The dividend is first rewritten like this:
42:by another polynomial of the same or lower
3766:
3752:
3744:
3658:Greatest common divisor of two polynomials
3536:
3523:
3503:
3494:
3470:
3468:
3452:
3443:
3429:
3420:
3413:
3382:
3366:
3353:
3329:
3308:
3306:
3257:
3241:
3229:
3162:then the equation of the tangent line at
3036:
3021:
2982:
2738:
2564:
2530:
2516:
2506:
2495:
2483:
2472:
2466:
2446:
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2421:
2379:
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2291:
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2172:
2138:
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2097:
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2057:
1962:
1944:
1935:
1919:
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1907:
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1819:
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1782:
1780:
1749:
1733:
1728:
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1649:
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1569:
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1516:
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1508:
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1212:
1196:
1189:
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1142:
1129:
1103:
1101:
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1082:
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1072:
1032:
1025:
1014:
987:
971:
955:
948:
926:
895:
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866:
844:
826:
802:
798:
796:
731:
715:
689:
658:
642:
629:
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561:
559:
489:
473:
460:
438:
420:
396:
392:
390:
325:
309:
303:
264:
230:
214:
208:
3103:Finding tangents to polynomial functions
2209:). Place the result (+x) below the bar.
3669:
2821:). Polynomial long division is thus an
1026:
3142:) is the remainder of the division of
61:, which starting from two polynomials
3504:
3469:
3453:
3430:
2237:as used and place the new remainder 3
1609:
1568:
1515:
1394:
1313:
1272:
1219:
1102:
1092:
988:
927:
803:
690:
566:
397:
16:Algorithm for division of polynomials
7:
3282:Begin by dividing the polynomial by
2716:≠ 0, polynomial division provides a
2657:This works equally well when degree(
2624:The algorithm can be represented in
2045:as used and place the new remainder
128:. These conditions uniquely define
3272:{\displaystyle y=x^{3}-12x^{2}-42.}
2676:is written on the left of the ")";
2017:). Place the result below the bar.
371:). Place the result above the bar (
349:{\displaystyle x^{3}-2x^{2}+0x-4.}
14:
3077:) to obtain a quotient of degree
2700:Euclidean division of polynomials
59:Euclidean division of polynomials
2704:For every pair of polynomials (
2582:
2540:
2539:
2538:
2537:
2536:
2465:
2464:
2456:
2445:
2430:
2429:
2428:
2427:
2391:
2355:
2354:
2353:
2352:
2351:
2284:
2283:
2272:
2257:
2256:
2255:
2179:
2178:
2148:
2147:
2146:
2145:
2144:
2079:
2078:
2063:
1972:
1971:
1970:
1969:
1968:
1914:
1913:
248:{\displaystyle x^{3}-2x^{2}-4,}
3356:
3044:
3030:
3011:
2999:
2996:
2984:
1858:
1852:
1829:
1823:
1810:
1785:
1777:
1765:
1428:
1132:
1001:
940:
869:
816:
703:
632:
579:
463:
410:
1:
3956:Horner's method of evaluation
3206:is a root of the polynomial.
3202:regardless of whether or not
3172:to the graph of the function
2949:can be divided out to obtain
2937:) are known, a linear factor
3632:Polynomial remainder theorem
3404:
2894:) is a polynomial of degree
1476:
1180:
917:
680:
511:
124:is lower than the degree of
3961:Polynomial identity testing
3737:89, November 2005: 466-467.
2921:Likewise, if several roots
4023:
3711:. READ BOOKS. p. 24.
3115:defined by the polynomial
2697:
18:
2910:is known to be a root of
1888:Polynomial short division
3123:) at a particular point
2825:for Euclidean division.
367:, which in this case is
199:Polynomial long division
32:polynomial long division
3946:Greatest common divisor
3620:cyclic redundancy check
3614:Cyclic redundancy check
2819:division transformation
2764:{\displaystyle A=BQ+R,}
4007:Division (mathematics)
3818:Quadratic function (2)
3590:
3273:
3069:can be divided out of
3063:
2969:can be divided out of
2765:
2646:degree(r) ≥ degree(d)
2591:
2400:
2188:
1996:
1867:
1691:
1357:
1055:
745:
524:
350:
282:
249:
3801:Constant function (0)
3591:
3274:
3113:graph of the function
3064:
2840:rational root theorem
2834:Factoring polynomials
2766:
2592:
2401:
2189:
1997:
1868:
1692:
1358:
1056:
746:
525:
351:
283:
250:
120:= 0 or the degree of
3934:Tools and algorithms
3854:Quintic function (5)
3842:Quartic function (4)
3779:polynomial functions
3734:Mathematical Gazette
3599:The tangent line is
3305:
3228:
3097:Abel–Ruffini theorem
2981:
2737:
2524:
2514:
2501:
2478:
2454:
2443:
2420:
2333:
2320:
2297:
2270:
2248:
2115:
2092:
2056:
1906:
1725:
1373:
1071:
795:
558:
389:
302:
281:{\displaystyle x-3,}
263:
207:
3864:Septic equation (7)
3859:Sextic equation (6)
3806:Linear function (1)
3707:S. Barnard (2008).
136:, which means that
3830:Cubic function (3)
3823:Quadratic equation
3637:Synthetic division
3586:
3584:
3561:
3508:
3478:
3466:
3457:
3434:
3269:
3093:quartic polynomial
3089:quintic polynomial
3059:
2815:Euclidean division
2761:
2692:Euclidean division
2587:
2585:
2554:
2396:
2394:
2369:
2184:
2182:
2162:
1992:
1990:
1986:
1894:mental calculation
1863:
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1111:
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946:
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741:
739:
709:
601:
520:
518:
432:
346:
278:
245:
52:synthetic division
21:synthetic division
3984:
3983:
3925:Quasi-homogeneous
3718:978-1-4437-3086-0
3524:
3414:
3407:
3361:
3352:
2531:
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2139:
1963:
1839:
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514:
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4014:
4002:Computer algebra
3847:Quartic equation
3768:
3761:
3754:
3745:
3738:
3729:
3723:
3722:
3704:
3698:
3697:
3696:
3695:
3674:
3648:Euclidean domain
3609:
3595:
3593:
3592:
3587:
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3562:
3557:
3541:
3540:
3509:
3499:
3498:
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3333:
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3201:
3186:
3171:
3161:
3133:
3083:
3068:
3066:
3065:
3060:
3040:
3026:
3025:
2968:
2948:
2885:
2866:) into the form
2846:of a polynomial
2770:
2768:
2767:
2762:
2596:
2594:
2593:
2588:
2586:
2569:
2568:
2555:
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2403:
2402:
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2370:
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2321:
2319:
2318:
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2271:
2269:
2268:
2193:
2191:
2190:
2185:
2183:
2177:
2176:
2163:
2158:
2127:
2116:
2114:
2113:
2093:
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2090:
2073:
2072:
2001:
1999:
1998:
1993:
1991:
1987:
1982:
1951:
1940:
1939:
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1813:
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3811:Linear equation
3781:
3772:
3742:
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3706:
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3701:
3693:
3691:
3685:
3682:Wayback Machine
3675:
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3600:
3583:
3582:
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3563:
3532:
3525:
3520:
3519:
3490:
3481:
3480:
3439:
3416:
3415:
3410:
3409:
3378:
3362:
3355:
3325:
3322:
3321:
3303:
3302:
3283:
3253:
3237:
3226:
3225:
3215:
3212:
3188:
3173:
3163:
3151:
3124:
3105:
3078:
3017:
2979:
2978:
2958:
2938:
2867:
2836:
2831:
2735:
2734:
2702:
2694:
2687:
2683:
2679:
2675:
2655:
2622:
2584:
2583:
2560:
2557:
2556:
2532:
2527:
2526:
2491:
2468:
2461:
2460:
2433:
2418:
2417:
2393:
2392:
2375:
2372:
2371:
2347:
2342:
2341:
2310:
2287:
2280:
2279:
2260:
2246:
2245:
2181:
2180:
2168:
2165:
2164:
2140:
2135:
2134:
2105:
2082:
2075:
2074:
2064:
2054:
2053:
1989:
1988:
1964:
1959:
1958:
1931:
1915:
1904:
1903:
1890:
1788:
1784:
1745:
1729:
1723:
1722:
1684:
1683:
1674:
1673:
1651:
1646:
1645:
1627:
1626:
1589:
1585:
1580:
1579:
1548:
1542:
1541:
1504:
1488:
1487:
1482:
1481:
1450:
1434:
1427:
1410:
1409:
1380:
1371:
1370:
1350:
1349:
1331:
1330:
1293:
1289:
1284:
1283:
1252:
1246:
1245:
1208:
1192:
1191:
1186:
1185:
1154:
1138:
1131:
1114:
1113:
1078:
1069:
1068:
1048:
1047:
1028:
1010:
984:
983:
967:
951:
950:
923:
922:
891:
875:
868:
851:
850:
840:
822:
793:
792:
759:
757:
738:
737:
727:
711:
686:
685:
654:
638:
631:
614:
613:
603:
585:
556:
555:
535:
517:
516:
485:
469:
462:
445:
444:
434:
416:
387:
386:
321:
305:
300:
299:
261:
260:
226:
210:
205:
204:
201:
196:
155:the polynomial
81:is not zero, a
77:) produces, if
38:for dividing a
24:
17:
12:
11:
5:
4020:
4018:
4010:
4009:
4004:
3999:
3989:
3988:
3982:
3981:
3979:
3978:
3973:
3968:
3963:
3958:
3953:
3948:
3943:
3937:
3935:
3931:
3930:
3928:
3927:
3922:
3917:
3912:
3907:
3902:
3897:
3892:
3887:
3882:
3876:
3874:
3870:
3869:
3867:
3866:
3861:
3856:
3851:
3850:
3849:
3839:
3838:
3837:
3835:Cubic equation
3827:
3826:
3825:
3815:
3814:
3813:
3803:
3798:
3792:
3790:
3783:
3782:
3773:
3771:
3770:
3763:
3756:
3748:
3740:
3739:
3724:
3717:
3709:Higher Algebra
3699:
3668:
3667:
3665:
3662:
3661:
3660:
3655:
3650:
3645:
3643:Ruffini's rule
3640:
3634:
3627:
3624:
3615:
3612:
3597:
3596:
3581:
3578:
3575:
3572:
3569:
3566:
3565:
3560:
3556:
3553:
3550:
3547:
3544:
3539:
3535:
3531:
3528:
3522:
3521:
3518:
3515:
3512:
3507:
3502:
3497:
3493:
3489:
3486:
3483:
3482:
3477:
3474:
3465:
3461:
3456:
3451:
3446:
3442:
3438:
3433:
3428:
3423:
3419:
3412:
3411:
3406:
3402:
3399:
3396:
3393:
3390:
3385:
3381:
3377:
3374:
3369:
3365:
3358:
3349:
3346:
3343:
3340:
3337:
3332:
3328:
3324:
3323:
3320:
3317:
3314:
3311:
3310:
3280:
3279:
3268:
3265:
3260:
3256:
3252:
3249:
3244:
3240:
3236:
3233:
3211:
3208:
3104:
3101:
3058:
3055:
3052:
3049:
3046:
3043:
3039:
3035:
3032:
3029:
3024:
3020:
3016:
3013:
3010:
3007:
3004:
3001:
2998:
2995:
2992:
2989:
2986:
2842:. If one root
2835:
2832:
2830:
2827:
2782:) < degree(
2772:
2771:
2760:
2757:
2754:
2751:
2748:
2745:
2742:
2698:Main article:
2693:
2690:
2685:
2681:
2677:
2673:
2661:) < degree(
2630:
2621:
2618:
2598:
2597:
2581:
2578:
2575:
2572:
2567:
2563:
2559:
2558:
2553:
2549:
2546:
2543:
2535:
2529:
2528:
2523:
2520:
2513:
2510:
2505:
2498:
2494:
2490:
2487:
2482:
2475:
2471:
2463:
2462:
2459:
2453:
2450:
2440:
2436:
2426:
2425:
2407:
2406:
2390:
2387:
2382:
2378:
2374:
2373:
2368:
2364:
2361:
2358:
2350:
2344:
2343:
2340:
2337:
2332:
2329:
2324:
2317:
2313:
2309:
2306:
2301:
2294:
2290:
2282:
2281:
2278:
2275:
2267:
2263:
2254:
2253:
2195:
2194:
2175:
2171:
2167:
2166:
2161:
2157:
2154:
2151:
2143:
2137:
2136:
2133:
2130:
2126:
2123:
2119:
2112:
2108:
2104:
2101:
2096:
2089:
2085:
2077:
2076:
2071:
2067:
2062:
2061:
2003:
2002:
1985:
1981:
1978:
1975:
1967:
1961:
1960:
1957:
1954:
1950:
1947:
1943:
1938:
1934:
1930:
1927:
1922:
1918:
1912:
1911:
1889:
1886:
1874:
1873:
1860:
1857:
1854:
1851:
1845:
1842:
1836:
1831:
1828:
1825:
1822:
1816:
1812:
1809:
1806:
1803:
1800:
1795:
1791:
1787:
1779:
1776:
1773:
1770:
1767:
1764:
1760:
1757:
1752:
1748:
1744:
1741:
1736:
1732:
1700:
1699:
1698:
1697:
1682:
1679:
1676:
1675:
1670:
1666:
1663:
1660:
1657:
1654:
1648:
1647:
1644:
1641:
1638:
1635:
1632:
1629:
1628:
1623:
1617:
1614:
1607:
1604:
1601:
1596:
1592:
1588:
1582:
1581:
1576:
1573:
1566:
1563:
1560:
1555:
1551:
1547:
1544:
1543:
1538:
1532:
1529:
1526:
1523:
1520:
1511:
1507:
1503:
1500:
1495:
1491:
1484:
1483:
1478:
1474:
1471:
1468:
1465:
1462:
1457:
1453:
1449:
1446:
1441:
1437:
1430:
1421:
1418:
1415:
1412:
1411:
1408:
1405:
1402:
1397:
1392:
1387:
1383:
1379:
1378:
1365:
1364:
1363:
1348:
1345:
1342:
1339:
1336:
1333:
1332:
1327:
1321:
1318:
1311:
1308:
1305:
1300:
1296:
1292:
1286:
1285:
1280:
1277:
1270:
1267:
1264:
1259:
1255:
1251:
1248:
1247:
1242:
1236:
1233:
1230:
1227:
1224:
1215:
1211:
1207:
1204:
1199:
1195:
1188:
1187:
1182:
1178:
1175:
1172:
1169:
1166:
1161:
1157:
1153:
1150:
1145:
1141:
1134:
1125:
1122:
1119:
1116:
1115:
1110:
1107:
1100:
1095:
1090:
1085:
1081:
1077:
1076:
1063:
1062:
1061:
1046:
1043:
1040:
1035:
1031:
1024:
1017:
1013:
1009:
1003:
997:
994:
991:
986:
985:
980:
974:
970:
966:
963:
958:
954:
942:
936:
933:
930:
925:
924:
919:
915:
912:
909:
906:
903:
898:
894:
890:
887:
882:
878:
871:
862:
859:
856:
853:
852:
847:
843:
837:
834:
829:
825:
818:
812:
809:
806:
801:
800:
753:
752:
751:
734:
730:
726:
723:
718:
714:
705:
699:
696:
693:
688:
687:
682:
678:
675:
672:
669:
666:
661:
657:
653:
650:
645:
641:
634:
625:
622:
619:
616:
615:
610:
606:
600:
597:
592:
588:
581:
575:
572:
569:
564:
563:
532:
531:
530:
513:
509:
506:
503:
500:
497:
492:
488:
484:
481:
476:
472:
465:
456:
453:
450:
447:
446:
441:
437:
431:
428:
423:
419:
412:
406:
403:
400:
395:
394:
357:
356:
345:
342:
339:
336:
333:
328:
324:
320:
317:
312:
308:
277:
274:
271:
268:
244:
241:
238:
233:
229:
225:
222:
217:
213:
200:
197:
195:
192:
153:if and only if
114:
113:
15:
13:
10:
9:
6:
4:
3:
2:
4019:
4008:
4005:
4003:
4000:
3998:
3995:
3994:
3992:
3977:
3976:Gröbner basis
3974:
3972:
3969:
3967:
3964:
3962:
3959:
3957:
3954:
3952:
3949:
3947:
3944:
3942:
3941:Factorization
3939:
3938:
3936:
3932:
3926:
3923:
3921:
3918:
3916:
3913:
3911:
3908:
3906:
3903:
3901:
3898:
3896:
3893:
3891:
3888:
3886:
3883:
3881:
3878:
3877:
3875:
3873:By properties
3871:
3865:
3862:
3860:
3857:
3855:
3852:
3848:
3845:
3844:
3843:
3840:
3836:
3833:
3832:
3831:
3828:
3824:
3821:
3820:
3819:
3816:
3812:
3809:
3808:
3807:
3804:
3802:
3799:
3797:
3794:
3793:
3791:
3789:
3784:
3780:
3776:
3769:
3764:
3762:
3757:
3755:
3750:
3749:
3746:
3736:
3735:
3728:
3725:
3720:
3714:
3710:
3703:
3700:
3690:
3689:
3683:
3679:
3673:
3670:
3663:
3659:
3656:
3654:
3653:Gröbner basis
3651:
3649:
3646:
3644:
3641:
3638:
3635:
3633:
3630:
3629:
3625:
3623:
3621:
3613:
3611:
3607:
3603:
3579:
3576:
3573:
3570:
3567:
3558:
3554:
3551:
3548:
3545:
3542:
3537:
3533:
3529:
3526:
3516:
3513:
3510:
3505:
3500:
3495:
3491:
3487:
3484:
3475:
3472:
3463:
3459:
3454:
3449:
3444:
3440:
3436:
3431:
3426:
3421:
3417:
3400:
3397:
3394:
3391:
3388:
3383:
3379:
3375:
3372:
3367:
3363:
3347:
3344:
3341:
3338:
3335:
3330:
3326:
3318:
3315:
3312:
3301:
3300:
3299:
3295:
3291:
3287:
3266:
3263:
3258:
3254:
3250:
3247:
3242:
3238:
3234:
3231:
3224:
3223:
3222:
3218:
3209:
3207:
3205:
3199:
3195:
3191:
3184:
3180:
3176:
3170:
3166:
3159:
3155:
3149:
3145:
3141:
3137:
3131:
3127:
3122:
3118:
3114:
3110:
3102:
3100:
3098:
3094:
3090:
3084:
3081:
3076:
3072:
3056:
3053:
3050:
3047:
3041:
3037:
3033:
3027:
3022:
3018:
3014:
3008:
3005:
3002:
2993:
2990:
2987:
2976:
2972:
2966:
2962:
2956:
2952:
2946:
2942:
2936:
2932:
2928:
2924:
2919:
2917:
2913:
2909:
2905:
2901:
2897:
2893:
2889:
2883:
2879:
2875:
2871:
2865:
2861:
2857:
2853:
2849:
2845:
2841:
2833:
2828:
2826:
2824:
2820:
2816:
2812:
2808:
2804:
2800:
2795:
2793:
2789:
2786:). Moreover (
2785:
2781:
2778:=0 or degree(
2777:
2758:
2755:
2752:
2749:
2746:
2743:
2740:
2733:
2732:
2731:
2729:
2726:
2722:
2719:
2715:
2711:
2707:
2701:
2696:
2691:
2689:
2670:
2668:
2664:
2660:
2653:
2649:
2645:
2641:
2637:
2633:
2629:
2627:
2619:
2617:
2615:
2611:
2607:
2603:
2579:
2576:
2573:
2570:
2565:
2561:
2551:
2547:
2544:
2541:
2533:
2521:
2518:
2511:
2508:
2503:
2496:
2492:
2488:
2485:
2480:
2473:
2469:
2457:
2451:
2448:
2438:
2434:
2416:
2415:
2414:
2412:
2388:
2385:
2380:
2376:
2366:
2362:
2359:
2356:
2348:
2338:
2335:
2330:
2327:
2322:
2315:
2311:
2307:
2304:
2299:
2292:
2288:
2276:
2273:
2265:
2261:
2244:
2243:
2242:
2240:
2236:
2232:
2228:
2224:
2220:
2216:
2212:
2208:
2204:
2200:
2173:
2169:
2159:
2155:
2152:
2149:
2141:
2131:
2128:
2124:
2121:
2117:
2110:
2106:
2102:
2099:
2094:
2087:
2083:
2069:
2065:
2052:
2051:
2050:
2048:
2044:
2040:
2036:
2032:
2028:
2024:
2020:
2016:
2012:
2008:
1983:
1979:
1976:
1973:
1965:
1955:
1952:
1948:
1945:
1941:
1936:
1932:
1928:
1925:
1920:
1916:
1902:
1901:
1900:
1897:
1895:
1887:
1885:
1883:
1879:
1878:long division
1855:
1849:
1843:
1840:
1834:
1826:
1820:
1814:
1807:
1804:
1801:
1798:
1793:
1789:
1774:
1771:
1768:
1762:
1758:
1755:
1750:
1746:
1742:
1739:
1734:
1730:
1721:
1720:
1719:
1717:
1713:
1709:
1705:
1680:
1677:
1668:
1664:
1661:
1658:
1655:
1652:
1642:
1639:
1636:
1633:
1630:
1621:
1615:
1612:
1605:
1602:
1599:
1594:
1590:
1586:
1574:
1571:
1564:
1561:
1558:
1553:
1549:
1545:
1536:
1530:
1527:
1524:
1521:
1518:
1509:
1505:
1501:
1498:
1493:
1489:
1472:
1469:
1466:
1463:
1460:
1455:
1451:
1447:
1444:
1439:
1435:
1419:
1416:
1413:
1406:
1403:
1400:
1395:
1390:
1385:
1381:
1369:
1368:
1366:
1346:
1343:
1340:
1337:
1334:
1325:
1319:
1316:
1309:
1306:
1303:
1298:
1294:
1290:
1278:
1275:
1268:
1265:
1262:
1257:
1253:
1249:
1240:
1234:
1231:
1228:
1225:
1222:
1213:
1209:
1205:
1202:
1197:
1193:
1176:
1173:
1170:
1167:
1164:
1159:
1155:
1151:
1148:
1143:
1139:
1123:
1120:
1117:
1108:
1105:
1098:
1093:
1088:
1083:
1079:
1067:
1066:
1064:
1044:
1041:
1038:
1033:
1029:
1022:
1015:
1011:
1007:
995:
992:
989:
978:
972:
968:
964:
961:
956:
952:
934:
931:
928:
913:
910:
907:
904:
901:
896:
892:
888:
885:
880:
876:
860:
857:
854:
845:
841:
835:
832:
827:
823:
810:
807:
804:
791:
790:
786:
782:
778:
774:
770:
766:
762:
754:
732:
728:
724:
721:
716:
712:
697:
694:
691:
676:
673:
670:
667:
664:
659:
655:
651:
648:
643:
639:
623:
620:
617:
608:
604:
598:
595:
590:
586:
573:
570:
567:
554:
553:
550:
546:
542:
538:
533:
507:
504:
501:
498:
495:
490:
486:
482:
479:
474:
470:
454:
451:
448:
439:
435:
429:
426:
421:
417:
404:
401:
398:
385:
384:
382:
378:
374:
370:
366:
362:
361:
360:
343:
340:
337:
334:
331:
326:
322:
318:
315:
310:
306:
298:
297:
296:
293:
291:
275:
272:
269:
266:
258:
242:
239:
236:
231:
227:
223:
220:
215:
211:
198:
193:
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:
3971:Discriminant
3950:
3890:Multivariate
3732:
3727:
3708:
3702:
3692:, retrieved
3687:
3678:Ghostarchive
3676:Archived at
3672:
3617:
3605:
3601:
3598:
3293:
3289:
3285:
3281:
3216:
3213:
3203:
3197:
3193:
3189:
3182:
3178:
3174:
3168:
3164:
3157:
3153:
3147:
3143:
3139:
3135:
3129:
3125:
3120:
3116:
3106:
3085:
3079:
3074:
3070:
2974:
2970:
2964:
2960:
2957:), and then
2954:
2950:
2944:
2940:
2934:
2930:
2929:, . . . of
2926:
2922:
2920:
2915:
2911:
2907:
2903:
2899:
2895:
2891:
2887:
2881:
2877:
2873:
2869:
2863:
2859:
2855:
2854:) of degree
2851:
2847:
2843:
2837:
2829:Applications
2818:
2814:
2810:
2806:
2802:
2798:
2796:
2791:
2787:
2783:
2779:
2775:
2773:
2727:
2724:
2720:
2717:
2713:
2712:) such that
2709:
2705:
2703:
2695:
2671:
2666:
2662:
2658:
2656:
2651:
2647:
2643:
2639:
2635:
2631:
2623:
2613:
2609:
2605:
2601:
2599:
2410:
2408:
2238:
2234:
2230:
2226:
2222:
2218:
2214:
2210:
2206:
2202:
2198:
2196:
2046:
2042:
2038:
2034:
2030:
2026:
2022:
2018:
2014:
2010:
2006:
2004:
1898:
1891:
1881:
1875:
1715:
1711:
1707:
1703:
1701:
784:
780:
776:
772:
768:
764:
760:
548:
544:
540:
536:
380:
376:
372:
368:
364:
358:
294:
289:
256:
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:
3997:Polynomials
3920:Homogeneous
3915:Square-free
3910:Irreducible
3775:Polynomials
2817:(sometimes
2774:and either
2730:such that
151:= 0 occurs
147:The result
116:and either
3991:Categories
3880:Univariate
3694:2019-12-10
3664:References
2813:is called
2626:pseudocode
2620:Pseudocode
2241:above it.
2049:above it.
97:such that
40:polynomial
3966:Resultant
3905:Trinomial
3885:Bivariate
3577:−
3568:−
3559:_
3552:−
3527:−
3514:−
3501:−
3485:−
3473:−
3464:_
3427:−
3405:¯
3398:−
3373:−
3336:−
3316:−
3264:−
3248:−
3028:−
3006:−
2991:−
2823:algorithm
2725:remainder
2552:_
2545:−
2534:÷
2519:−
2486:−
2367:_
2360:−
2349:÷
2336:−
2305:−
2160:_
2153:−
2142:÷
2129:−
2100:−
2041:. Mark −2
1984:_
1977:−
1966:÷
1953:−
1926:−
1844:⏟
1815:⏟
1772:−
1756:−
1740:−
1669:_
1662:−
1640:−
1622:_
1613:−
1600:−
1572:−
1537:_
1528:−
1499:−
1477:¯
1470:−
1445:−
1417:−
1344:−
1326:_
1317:−
1304:−
1276:−
1241:_
1232:−
1203:−
1181:¯
1174:−
1149:−
1121:−
993:−
979:_
962:−
932:−
918:¯
911:−
886:−
858:−
833:−
808:−
722:−
695:−
681:¯
674:−
649:−
621:−
596:−
571:−
512:¯
505:−
480:−
452:−
427:−
402:−
341:−
316:−
270:−
237:−
221:−
92:remainder
36:algorithm
3951:Division
3900:Binomial
3895:Monomial
3680:and the
3626:See also
2718:quotient
2632:function
2233:. Mark 0
257:dividend
84:quotient
67:dividend
3288:− 1) =
3210:Example
3111:to the
3109:tangent
2654:(q, r)
543:− 3) =
290:divisor
194:Example
75:divisor
28:algebra
3788:degree
3715:
3360:
3351:
2898:− 1.
2886:where
2723:and a
2652:return
2642:r ≠ 0
2634:n / d
1432:
1423:
1136:
1127:
1005:
999:
944:
938:
873:
864:
820:
814:
775:) = −2
707:
701:
636:
627:
583:
577:
467:
458:
414:
408:
165:factor
90:and a
69:) and
44:degree
34:is an
3604:= −21
3150:) by
2805:from
2640:while
2229:) = 3
2225:− (−3
2033:− (−3
767:) − (
259:, by
163:as a
73:(the
65:(the
3777:and
3713:ISBN
3608:− 32
3082:− 2.
2809:and
2801:and
2037:) =
1876:The
288:the
255:the
182:by (
169:root
159:has
140:and
132:and
3786:By
3296:+ 1
3292:− 2
3267:42.
3219:= 1
3187:is
3134:If
2669:).
2644:and
2616:).
1718:).
779:+ 3
771:− 3
763:− 2
552:).
547:− 3
539:· (
383:).
190:).
174:of
26:In
3993::
3684::
3618:A
3610:.
3580:32
3571:21
3555:10
3546:20
3530:10
3517:42
3506:01
3488:10
3476:42
3401:42
3376:12
3319:10
3298::
3251:12
3221::
3200:),
3192:=
3177:=
3167:=
3160:),
3156:–
3128:=
3099:.
2963:−
2943:−
2925:,
2872:−
2790:,
2708:,
2688:.
2648:do
2636:is
2205:=
2201:÷
2013:=
2009:÷
783:=
379:=
375:÷
344:4.
292:.
108:+
106:BQ
104:=
30:,
3767:e
3760:t
3753:v
3721:.
3606:x
3602:y
3574:x
3549:x
3543:+
3538:2
3534:x
3511:x
3496:2
3492:x
3460:x
3455:1
3450:+
3445:2
3441:x
3437:2
3432:0
3422:3
3418:x
3395:x
3392:0
3389:+
3384:2
3380:x
3368:3
3364:x
3357:)
3348:1
3345:+
3342:x
3339:2
3331:2
3327:x
3313:x
3294:x
3290:x
3286:x
3284:(
3259:2
3255:x
3243:3
3239:x
3235:=
3232:y
3217:x
3204:r
3198:x
3196:(
3194:R
3190:y
3185:)
3183:x
3181:(
3179:P
3175:y
3169:r
3165:x
3158:r
3154:x
3152:(
3148:x
3146:(
3144:P
3140:x
3138:(
3136:R
3132:.
3130:r
3126:x
3121:x
3119:(
3117:P
3080:n
3075:x
3073:(
3071:P
3057:s
3054:r
3051:+
3048:x
3045:)
3042:s
3038:+
3034:r
3031:(
3023:2
3019:x
3015:=
3012:)
3009:s
3003:x
3000:(
2997:)
2994:r
2988:x
2985:(
2975:x
2973:(
2971:Q
2967:)
2965:s
2961:x
2959:(
2955:x
2953:(
2951:Q
2947:)
2945:r
2941:x
2939:(
2935:x
2933:(
2931:P
2927:s
2923:r
2916:x
2914:(
2912:P
2908:r
2904:x
2902:(
2900:Q
2896:n
2892:x
2890:(
2888:Q
2884:)
2882:x
2880:(
2878:Q
2876:)
2874:r
2870:x
2868:(
2864:x
2862:(
2860:P
2856:n
2852:x
2850:(
2848:P
2844:r
2811:B
2807:A
2803:R
2799:Q
2792:R
2788:Q
2784:B
2780:R
2776:R
2759:,
2756:R
2753:+
2750:Q
2747:B
2744:=
2741:A
2728:R
2721:Q
2714:B
2710:B
2706:A
2686:r
2682:t
2678:q
2674:d
2667:n
2663:d
2659:n
2614:x
2612:(
2610:r
2606:x
2604:(
2602:q
2580:3
2577:+
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2571:+
2566:2
2562:x
2548:3
2542:x
2522:4
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2509:0
2504:+
2497:2
2493:x
2489:2
2481:+
2474:3
2470:x
2458:5
2452:x
2449:3
2439:2
2435:x
2411:x
2389:x
2386:+
2381:2
2377:x
2363:3
2357:x
2339:4
2331:x
2328:0
2323:+
2316:2
2312:x
2308:2
2300:+
2293:3
2289:x
2277:x
2274:3
2266:2
2262:x
2239:x
2235:x
2231:x
2227:x
2223:x
2219:x
2215:x
2211:x
2207:x
2203:x
2199:x
2174:2
2170:x
2156:3
2150:x
2132:4
2125:x
2122:0
2118:+
2111:2
2107:x
2103:2
2095:+
2088:3
2084:x
2070:2
2066:x
2047:x
2043:x
2039:x
2035:x
2031:x
2027:x
2023:x
2019:x
2015:x
2011:x
2007:x
1980:3
1974:x
1956:4
1949:x
1946:0
1942:+
1937:2
1933:x
1929:2
1921:3
1917:x
1882:x
1859:)
1856:x
1853:(
1850:r
1841:5
1835:+
1830:)
1827:x
1824:(
1821:q
1811:)
1808:3
1805:+
1802:x
1799:+
1794:2
1790:x
1786:(
1778:)
1775:3
1769:x
1766:(
1763:=
1759:4
1751:2
1747:x
1743:2
1735:3
1731:x
1716:x
1714:(
1712:r
1708:x
1706:(
1704:q
1681:5
1678:+
1665:9
1659:x
1656:3
1653:+
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1634:3
1631:+
1616:4
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1591:x
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1562:0
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1546:+
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1525:x
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1506:x
1502:3
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1467:x
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1452:x
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1429:)
1420:3
1414:x
1407:3
1404:+
1401:x
1396:1
1391:+
1386:2
1382:x
1347:4
1341:x
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1320:4
1310:x
1307:3
1299:2
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1291:+
1279:4
1269:x
1266:0
1263:+
1258:2
1254:x
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1223:+
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1198:3
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1177:4
1171:x
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1165:+
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1140:x
1133:)
1124:3
1118:x
1109:3
1106:+
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1094:1
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1080:x
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1008:0
1002:)
996:3
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969:x
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941:)
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902:+
897:2
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870:)
861:3
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846:2
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756:(
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327:2
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276:,
273:3
267:x
243:,
240:4
232:2
228:x
224:2
216:3
212:x
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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