43:
990:); likewise, a real polynomial of even degree must have an even number of real roots. Consequently, real odd polynomials must have at least one real root (because the smallest odd whole number is 1), whereas even polynomials may have none. This principle can be proven by reference to the
782:
225:
968:. In other words, a "zero of a function" is precisely a "solution of the equation obtained by equating the function to 0", and the study of zeros of functions is exactly the same as the study of solutions of equations.
654:
1941:
262:
1409:
1161:
1887:
1745:
1716:
1644:
1611:
1208:
84:
1559:
943:
667:
509:
1821:
1238:
847:
149:
387:
1315:
1358:
1848:
1789:
1769:
1683:
1473:
1453:
1433:
1335:
1289:
1258:
1050:
1030:
966:
905:
868:
804:
564:
470:
450:
430:
410:
358:
334:
314:
104:
158:
998:, the function value must cross zero, in the process of changing from negative to positive or vice versa (which always happens for odd functions).
987:
948:
by regrouping all the terms in the left-hand side. It follows that the solutions of such an equation are exactly the zeros of the function
538:, and that the number of roots and the degree are equal when one considers the complex roots (or more generally, the roots in an
17:
2062:
2046:
1824:
1052:
complex roots, counted with their multiplicities. The non-real roots of polynomials with real coefficients come in
2123:
2118:
569:
1007:
977:
539:
527:
1892:
230:
1380:
1132:
1857:
991:
543:
2000:
1721:
1692:
1620:
1587:
1412:
2038:
2028:
1962:
1957:
1174:
1126:
1095:
1069:
983:
535:
473:
294:
31:
1658:
1573:
885:
57:
51:
1505:
913:
479:
1794:
1492:
1217:
1164:
820:
337:
109:
363:
8:
2128:
1952:
1562:
1294:
1083:
1073:
1057:
995:
807:
523:
1340:
1967:
1833:
1774:
1754:
1668:
1484:
1458:
1438:
1418:
1320:
1274:
1243:
1103:
1099:
1091:
1035:
1015:
951:
890:
853:
789:
549:
455:
435:
415:
395:
343:
319:
299:
89:
2091:
2042:
1488:
1053:
1496:
1367:
1077:
1977:
1686:
1651:
1647:
1614:
1500:
777:{\displaystyle f(2)=2^{2}-5\times 2+6=0{\text{ and }}f(3)=3^{2}-5\times 3+6=0.}
290:
220:{\displaystyle -{\tfrac {3\pi }{2}},\;-{\tfrac {\pi }{2}},\;{\tfrac {\pi }{2}}}
511:. A "zero" of a function is thus an input value that produces an output of 0.
2112:
2034:
1972:
1748:
1211:
1168:
1087:
1060:
relate the coefficients of a polynomial to sums and products of its roots.
30:"Root of a function" redirects here. For a half iterate of a function, see
2094:
1012:
The fundamental theorem of algebra states that every polynomial of degree
786:
If the function maps real numbers to real numbers, then its zeros are the
286:
274:
2030:
Algebra and
Trigonometry: Functions and Applications, Teacher's Edition
1581:
1363:
531:
519:
2099:
1268:
1662:
1577:
1264:
1115:
881:
1851:
811:
42:
1094:). However, some polynomial functions, including all those of
1167:(or, more generally, a function taking values in some
1114:"Zero set" redirects here. For the musical album, see
1098:
no greater than 4, can have all their roots expressed
235:
206:
190:
166:
1895:
1860:
1836:
1797:
1777:
1757:
1724:
1695:
1671:
1623:
1590:
1508:
1461:
1441:
1421:
1383:
1343:
1323:
1297:
1277:
1246:
1220:
1177:
1135:
1038:
1018:
954:
916:
893:
856:
823:
792:
670:
572:
552:
482:
458:
438:
418:
398:
366:
346:
322:
302:
233:
161:
112:
92:
60:
1565:. In this context, a zero set is sometimes called a
2063:"Roots and zeros (Algebra 2, Polynomial functions)"
1935:
1881:
1842:
1815:
1783:
1763:
1739:
1710:
1677:
1638:
1605:
1553:
1467:
1447:
1427:
1403:
1352:
1329:
1309:
1283:
1252:
1232:
1202:
1155:
1044:
1024:
960:
937:
899:
862:
841:
798:
776:
648:
558:
503:
464:
444:
424:
404:
381:
352:
328:
308:
256:
219:
143:
98:
78:
1001:
2110:
1129:is the set of all its zeros. More precisely, if
1086:, frequently requires the use of specialised or
1499:of the zero sets of several polynomials, in a
1102:in terms of their coefficients (for more, see
2089:
2033:(Classics ed.). Upper Saddle River, NJ:
1665:. An important special case is the case that
1918:
1911:
1889:is the zero set of the real-valued function
1227:
1221:
534:has a number of roots at most equal to its
2022:
2020:
1661:, zero sets are frequently used to define
1082:Computing roots of functions, for example
986:has an odd number of real roots (counting
875:
649:{\displaystyle f(x)=x^{2}-5x+6=(x-2)(x-3)}
204:
185:
1863:
1727:
1698:
1626:
1593:
1397:
1149:
2026:
1936:{\displaystyle f(x)=\Vert x\Vert ^{2}-1}
2017:
2001:"Algebra - Zeroes/Roots of Polynomials"
1491:is through zero sets. Specifically, an
817:. An alternative name for such a point
14:
2111:
656:has the two roots (or zeros) that are
2090:
1121:In various areas of mathematics, the
806:-coordinates of the points where its
1995:
1993:
257:{\displaystyle {\tfrac {3\pi }{2}},}
27:Point where function's value is zero
1404:{\displaystyle f:X\to \mathbb {R} }
1156:{\displaystyle f:X\to \mathbb {R} }
971:
24:
2083:
1882:{\displaystyle \mathbb {R} ^{m+1}}
1791:is a smooth manifold of dimension
1063:
25:
2140:
1990:
1263:Under the same hypothesis on the
994:: since polynomial functions are
1740:{\displaystyle \mathbb {R} ^{n}}
1711:{\displaystyle \mathbb {R} ^{p}}
1639:{\displaystyle \mathbb {R} ^{n}}
1606:{\displaystyle \mathbb {R} ^{n}}
1291:is the zero set of the function
41:
1478:
522:is a zero of the corresponding
2055:
1905:
1899:
1393:
1197:
1191:
1145:
1008:Fundamental theorem of algebra
1002:Fundamental theorem of algebra
978:Properties of polynomial roots
926:
920:
836:
824:
734:
728:
680:
674:
643:
631:
628:
616:
582:
576:
546:. For example, the polynomial
540:algebraically closed extension
528:fundamental theorem of algebra
492:
486:
376:
370:
73:
67:
13:
1:
1983:
1487:, the first definition of an
982:Every real polynomial of odd
7:
1946:
1109:
10:
2145:
2027:Foerster, Paul A. (2006).
1113:
1067:
1005:
992:intermediate value theorem
975:
566:of degree two, defined by
432:attains the value of 0 at
29:
1203:{\displaystyle f^{-1}(0)}
281:(also sometimes called a
530:shows that any non-zero
412:; that is, the function
54:A graph of the function
2005:tutorial.math.lamar.edu
1771:, then the zero set of
876:Solution of an equation
79:{\displaystyle \cos(x)}
2124:Functions and mappings
2119:Elementary mathematics
1958:Root-finding algorithm
1937:
1883:
1844:
1830:For example, the unit
1817:
1785:
1765:
1741:
1712:
1679:
1646:. This extends to any
1640:
1607:
1555:
1554:{\displaystyle k\left}
1469:
1449:
1429:
1405:
1354:
1331:
1311:
1285:
1254:
1234:
1204:
1157:
1070:Root-finding algorithm
1046:
1026:
962:
939:
938:{\displaystyle f(x)=0}
901:
864:
849:in this context is an
843:
800:
778:
650:
560:
505:
504:{\displaystyle f(x)=0}
466:
446:
426:
406:
383:
354:
330:
310:
295:vector-valued function
258:
221:
145:
100:
80:
32:Functional square root
1938:
1884:
1845:
1825:regular value theorem
1818:
1816:{\displaystyle m=p-n}
1786:
1766:
1742:
1713:
1680:
1659:differential geometry
1641:
1613:is the zero set of a
1608:
1556:
1470:
1450:
1435:(i.e., the subset of
1430:
1406:
1366:is also known as its
1355:
1332:
1312:
1286:
1255:
1235:
1233:{\displaystyle \{0\}}
1205:
1158:
1047:
1027:
963:
940:
902:
865:
844:
842:{\displaystyle (x,0)}
801:
779:
651:
561:
542:) counted with their
506:
467:
447:
427:
407:
384:
355:
331:
311:
259:
222:
146:
144:{\displaystyle \left}
101:
81:
1893:
1858:
1834:
1795:
1775:
1755:
1722:
1693:
1669:
1621:
1588:
1506:
1493:affine algebraic set
1459:
1439:
1419:
1381:
1341:
1321:
1295:
1275:
1244:
1218:
1175:
1165:real-valued function
1133:
1084:polynomial functions
1036:
1016:
952:
914:
907:may be rewritten as
891:
854:
821:
790:
668:
570:
550:
480:
456:
436:
416:
396:
382:{\displaystyle f(x)}
364:
344:
320:
300:
231:
159:
110:
90:
58:
18:Root of a polynomial
1963:Sendov's conjecture
1415:of the zero set of
1337:in the codomain of
1310:{\displaystyle f-c}
1267:of the function, a
1171:), its zero set is
1074:Real-root isolation
524:polynomial function
452:, or equivalently,
2092:Weisstein, Eric W.
1968:Vanish at infinity
1933:
1879:
1840:
1813:
1781:
1761:
1737:
1708:
1675:
1650:as a corollary of
1636:
1617:defined on all of
1603:
1551:
1485:algebraic geometry
1465:
1445:
1425:
1401:
1362:The zero set of a
1353:{\displaystyle f.}
1350:
1327:
1307:
1281:
1250:
1230:
1200:
1153:
1104:algebraic solution
1090:techniques (e.g.,
1042:
1022:
958:
935:
897:
860:
839:
796:
774:
646:
556:
501:
462:
442:
422:
402:
379:
350:
326:
306:
254:
249:
217:
215:
199:
180:
141:
96:
76:
1843:{\displaystyle m}
1784:{\displaystyle f}
1764:{\displaystyle f}
1678:{\displaystyle f}
1489:algebraic variety
1468:{\displaystyle f}
1448:{\displaystyle X}
1428:{\displaystyle f}
1330:{\displaystyle c}
1284:{\displaystyle f}
1253:{\displaystyle X}
1045:{\displaystyle n}
1025:{\displaystyle n}
961:{\displaystyle f}
900:{\displaystyle x}
863:{\displaystyle x}
799:{\displaystyle x}
723:
559:{\displaystyle f}
465:{\displaystyle x}
445:{\displaystyle x}
425:{\displaystyle f}
405:{\displaystyle x}
353:{\displaystyle f}
329:{\displaystyle x}
309:{\displaystyle f}
248:
214:
198:
179:
99:{\displaystyle x}
16:(Redirected from
2136:
2105:
2104:
2077:
2076:
2074:
2073:
2059:
2053:
2052:
2024:
2015:
2014:
2012:
2011:
1997:
1953:Marden's theorem
1942:
1940:
1939:
1934:
1926:
1925:
1888:
1886:
1885:
1880:
1878:
1877:
1866:
1849:
1847:
1846:
1841:
1822:
1820:
1819:
1814:
1790:
1788:
1787:
1782:
1770:
1768:
1767:
1762:
1746:
1744:
1743:
1738:
1736:
1735:
1730:
1717:
1715:
1714:
1709:
1707:
1706:
1701:
1684:
1682:
1681:
1676:
1645:
1643:
1642:
1637:
1635:
1634:
1629:
1612:
1610:
1609:
1604:
1602:
1601:
1596:
1560:
1558:
1557:
1552:
1550:
1546:
1545:
1544:
1526:
1525:
1474:
1472:
1471:
1466:
1454:
1452:
1451:
1446:
1434:
1432:
1431:
1426:
1410:
1408:
1407:
1402:
1400:
1377:of the function
1359:
1357:
1356:
1351:
1336:
1334:
1333:
1328:
1316:
1314:
1313:
1308:
1290:
1288:
1287:
1282:
1259:
1257:
1256:
1251:
1239:
1237:
1236:
1231:
1209:
1207:
1206:
1201:
1190:
1189:
1162:
1160:
1159:
1154:
1152:
1078:Equation solving
1058:Vieta's formulas
1051:
1049:
1048:
1043:
1031:
1029:
1028:
1023:
972:Polynomial roots
967:
965:
964:
959:
944:
942:
941:
936:
906:
904:
903:
898:
869:
867:
866:
861:
848:
846:
845:
840:
805:
803:
802:
797:
783:
781:
780:
775:
749:
748:
724:
721:
695:
694:
655:
653:
652:
647:
597:
596:
565:
563:
562:
557:
510:
508:
507:
502:
476:to the equation
471:
469:
468:
463:
451:
449:
448:
443:
431:
429:
428:
423:
411:
409:
408:
403:
388:
386:
385:
380:
359:
357:
356:
351:
335:
333:
332:
327:
315:
313:
312:
307:
293:-, or generally
267:
263:
261:
260:
255:
250:
244:
236:
226:
224:
223:
218:
216:
207:
200:
191:
181:
175:
167:
150:
148:
147:
142:
140:
136:
105:
103:
102:
97:
85:
83:
82:
77:
45:
21:
2144:
2143:
2139:
2138:
2137:
2135:
2134:
2133:
2109:
2108:
2086:
2084:Further reading
2081:
2080:
2071:
2069:
2061:
2060:
2056:
2049:
2025:
2018:
2009:
2007:
1999:
1998:
1991:
1986:
1978:Zeros and poles
1949:
1921:
1917:
1894:
1891:
1890:
1867:
1862:
1861:
1859:
1856:
1855:
1835:
1832:
1831:
1796:
1793:
1792:
1776:
1773:
1772:
1756:
1753:
1752:
1747:. If zero is a
1731:
1726:
1725:
1723:
1720:
1719:
1702:
1697:
1696:
1694:
1691:
1690:
1687:smooth function
1670:
1667:
1666:
1652:paracompactness
1648:smooth manifold
1630:
1625:
1624:
1622:
1619:
1618:
1615:smooth function
1597:
1592:
1591:
1589:
1586:
1585:
1540:
1536:
1521:
1517:
1516:
1512:
1507:
1504:
1503:
1501:polynomial ring
1481:
1460:
1457:
1456:
1440:
1437:
1436:
1420:
1417:
1416:
1396:
1382:
1379:
1378:
1342:
1339:
1338:
1322:
1319:
1318:
1296:
1293:
1292:
1276:
1273:
1272:
1245:
1242:
1241:
1219:
1216:
1215:
1182:
1178:
1176:
1173:
1172:
1148:
1134:
1131:
1130:
1119:
1112:
1092:Newton's method
1080:
1068:Main articles:
1066:
1064:Computing roots
1037:
1034:
1033:
1017:
1014:
1013:
1010:
1004:
980:
974:
953:
950:
949:
915:
912:
911:
892:
889:
888:
878:
855:
852:
851:
822:
819:
818:
791:
788:
787:
744:
740:
722: and
720:
690:
686:
669:
666:
665:
592:
588:
571:
568:
567:
551:
548:
547:
481:
478:
477:
457:
454:
453:
437:
434:
433:
417:
414:
413:
397:
394:
393:
365:
362:
361:
345:
342:
341:
321:
318:
317:
301:
298:
297:
271:
270:
269:
265:
237:
234:
232:
229:
228:
205:
189:
168:
165:
160:
157:
156:
117:
113:
111:
108:
107:
91:
88:
87:
59:
56:
55:
53:
48:
47:
46:
35:
28:
23:
22:
15:
12:
11:
5:
2142:
2132:
2131:
2126:
2121:
2107:
2106:
2085:
2082:
2079:
2078:
2054:
2047:
2016:
1988:
1987:
1985:
1982:
1981:
1980:
1975:
1970:
1965:
1960:
1955:
1948:
1945:
1932:
1929:
1924:
1920:
1916:
1913:
1910:
1907:
1904:
1901:
1898:
1876:
1873:
1870:
1865:
1839:
1812:
1809:
1806:
1803:
1800:
1780:
1760:
1734:
1729:
1705:
1700:
1674:
1633:
1628:
1600:
1595:
1549:
1543:
1539:
1535:
1532:
1529:
1524:
1520:
1515:
1511:
1480:
1477:
1475:is nonzero).
1464:
1444:
1424:
1399:
1395:
1392:
1389:
1386:
1349:
1346:
1326:
1306:
1303:
1300:
1280:
1271:of a function
1249:
1229:
1226:
1223:
1199:
1196:
1193:
1188:
1185:
1181:
1169:additive group
1151:
1147:
1144:
1141:
1138:
1111:
1108:
1065:
1062:
1041:
1021:
1006:Main article:
1003:
1000:
988:multiplicities
976:Main article:
973:
970:
957:
946:
945:
934:
931:
928:
925:
922:
919:
896:
877:
874:
859:
838:
835:
832:
829:
826:
795:
773:
770:
767:
764:
761:
758:
755:
752:
747:
743:
739:
736:
733:
730:
727:
719:
716:
713:
710:
707:
704:
701:
698:
693:
689:
685:
682:
679:
676:
673:
645:
642:
639:
636:
633:
630:
627:
624:
621:
618:
615:
612:
609:
606:
603:
600:
595:
591:
587:
584:
581:
578:
575:
555:
544:multiplicities
500:
497:
494:
491:
488:
485:
461:
441:
421:
401:
378:
375:
372:
369:
349:
325:
316:, is a member
305:
253:
247:
243:
240:
213:
210:
203:
197:
194:
188:
184:
178:
174:
171:
164:
139:
135:
132:
129:
126:
123:
120:
116:
95:
75:
72:
69:
66:
63:
50:
49:
40:
39:
38:
37:
36:
26:
9:
6:
4:
3:
2:
2141:
2130:
2127:
2125:
2122:
2120:
2117:
2116:
2114:
2102:
2101:
2096:
2093:
2088:
2087:
2068:
2064:
2058:
2050:
2048:0-13-165711-9
2044:
2040:
2036:
2035:Prentice Hall
2032:
2031:
2023:
2021:
2006:
2002:
1996:
1994:
1989:
1979:
1976:
1974:
1973:Zero crossing
1971:
1969:
1966:
1964:
1961:
1959:
1956:
1954:
1951:
1950:
1944:
1930:
1927:
1922:
1914:
1908:
1902:
1896:
1874:
1871:
1868:
1853:
1837:
1828:
1826:
1810:
1807:
1804:
1801:
1798:
1778:
1758:
1750:
1749:regular value
1732:
1703:
1688:
1672:
1664:
1660:
1655:
1653:
1649:
1631:
1616:
1598:
1583:
1582:closed subset
1579:
1575:
1570:
1568:
1564:
1547:
1541:
1537:
1533:
1530:
1527:
1522:
1518:
1513:
1509:
1502:
1498:
1494:
1490:
1486:
1476:
1462:
1442:
1422:
1414:
1390:
1387:
1384:
1376:
1371:
1369:
1365:
1360:
1347:
1344:
1324:
1304:
1301:
1298:
1278:
1270:
1266:
1261:
1247:
1224:
1213:
1212:inverse image
1194:
1186:
1183:
1179:
1170:
1166:
1142:
1139:
1136:
1128:
1124:
1117:
1107:
1105:
1101:
1100:algebraically
1097:
1093:
1089:
1088:approximation
1085:
1079:
1075:
1071:
1061:
1059:
1055:
1039:
1019:
1009:
999:
997:
993:
989:
985:
979:
969:
955:
932:
929:
923:
917:
910:
909:
908:
894:
887:
883:
873:
871:
857:
833:
830:
827:
816:
814:
809:
793:
784:
771:
768:
765:
762:
759:
756:
753:
750:
745:
741:
737:
731:
725:
717:
714:
711:
708:
705:
702:
699:
696:
691:
687:
683:
677:
671:
663:
659:
640:
637:
634:
625:
622:
619:
613:
610:
607:
604:
601:
598:
593:
589:
585:
579:
573:
553:
545:
541:
537:
533:
529:
525:
521:
517:
512:
498:
495:
489:
483:
475:
459:
439:
419:
399:
391:
373:
367:
347:
339:
323:
303:
296:
292:
288:
284:
280:
276:
251:
245:
241:
238:
211:
208:
201:
195:
192:
186:
182:
176:
172:
169:
162:
154:
137:
133:
130:
127:
124:
121:
118:
114:
93:
70:
64:
61:
52:
44:
33:
19:
2098:
2070:. Retrieved
2066:
2057:
2029:
2008:. Retrieved
2004:
1829:
1656:
1571:
1566:
1497:intersection
1482:
1479:Applications
1374:
1372:
1361:
1262:
1122:
1120:
1081:
1011:
981:
947:
879:
850:
812:
785:
661:
657:
515:
513:
389:
282:
278:
272:
152:
275:mathematics
2129:0 (number)
2113:Categories
2072:2019-12-15
2067:Mathplanet
2037:. p.
2010:2019-12-15
1984:References
1567:zero locus
1413:complement
1375:cozero set
1364:linear map
996:continuous
870:-intercept
810:meets the
532:polynomial
520:polynomial
360:such that
264:marked in
2100:MathWorld
1928:−
1919:‖
1912:‖
1808:−
1663:manifolds
1531:…
1455:on which
1394:→
1317:for some
1302:−
1269:level set
1184:−
1146:→
1054:conjugate
757:×
751:−
703:×
697:−
638:−
623:−
599:−
242:π
209:π
193:π
187:−
173:π
163:−
134:π
125:π
119:−
65:
1947:See also
1578:geometry
1574:analysis
1265:codomain
1127:function
1123:zero set
1116:Zero Set
1110:Zero set
882:equation
474:solution
390:vanishes
1823:by the
1561:over a
1495:is the
1411:is the
1056:pairs.
886:unknown
884:in the
336:of the
291:complex
285:) of a
151:, with
2095:"Root"
2045:
1852:sphere
1580:, any
1368:kernel
1210:, the
1096:degree
1076:, and
984:degree
880:Every
536:degree
526:. The
338:domain
227:, and
1689:from
1685:is a
1563:field
1163:is a
1125:of a
815:-axis
808:graph
518:of a
472:is a
153:zeros
2043:ISBN
1576:and
1373:The
1032:has
660:and
516:root
287:real
283:root
279:zero
277:, a
86:for
2039:535
1854:in
1751:of
1718:to
1657:In
1654:.
1584:of
1572:In
1483:In
1260:.
1240:in
1214:of
1106:).
392:at
340:of
289:-,
273:In
266:red
155:at
106:in
62:cos
2115::
2097:.
2065:.
2041:.
2019:^
2003:.
1992:^
1943:.
1827:.
1569:.
1370:.
1072:,
872:.
772:0.
664:.
514:A
268:.
2103:.
2075:.
2051:.
2013:.
1931:1
1923:2
1915:x
1909:=
1906:)
1903:x
1900:(
1897:f
1875:1
1872:+
1869:m
1864:R
1850:-
1838:m
1811:n
1805:p
1802:=
1799:m
1779:f
1759:f
1733:n
1728:R
1704:p
1699:R
1673:f
1632:n
1627:R
1599:n
1594:R
1548:]
1542:n
1538:x
1534:,
1528:,
1523:1
1519:x
1514:[
1510:k
1463:f
1443:X
1423:f
1398:R
1391:X
1388::
1385:f
1348:.
1345:f
1325:c
1305:c
1299:f
1279:f
1248:X
1228:}
1225:0
1222:{
1198:)
1195:0
1192:(
1187:1
1180:f
1150:R
1143:X
1140::
1137:f
1118:.
1040:n
1020:n
956:f
933:0
930:=
927:)
924:x
921:(
918:f
895:x
858:x
837:)
834:0
831:,
828:x
825:(
813:x
794:x
769:=
766:6
763:+
760:3
754:5
746:2
742:3
738:=
735:)
732:3
729:(
726:f
718:0
715:=
712:6
709:+
706:2
700:5
692:2
688:2
684:=
681:)
678:2
675:(
672:f
662:3
658:2
644:)
641:3
635:x
632:(
629:)
626:2
620:x
617:(
614:=
611:6
608:+
605:x
602:5
594:2
590:x
586:=
583:)
580:x
577:(
574:f
554:f
499:0
496:=
493:)
490:x
487:(
484:f
460:x
440:x
420:f
400:x
377:)
374:x
371:(
368:f
348:f
324:x
304:f
252:,
246:2
239:3
212:2
202:,
196:2
183:,
177:2
170:3
138:]
131:2
128:,
122:2
115:[
94:x
74:)
71:x
68:(
34:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
