540.sagews

#1.Zadatak: Odredite polinom trećeg stupnja koji prolazi točkama (1,3), (2,16), (-1,1) i (3,45).

R.<x>=QQ[]
f=R.lagrange_polynomial([(1,3),(2,16),(-1,1),(3,45)])

f

x^{3} + 2 x^{2}

#grafički prikaz dobivenog polinoma i zadanih točaka

plot(f,-2,3.5)+point(((1,3),(2,16),(-1,1),(3,45)),rgbcolor=(1,0,0),pointsize=30,faceted=True)

plot(f,-2,3.5)+point(((1,3),(2,16),(-1,1),(3,45)),rgbcolor=(0,0,1),pointsize=30,faceted=True)

#ili možemo koristiti moćan paket matplotlib

import pylab

import numpy

x=numpy.arange(-2.0,3.5,0.01)

y=f(x)

tocke_x = [1,2,-1,3]
tocke_y = [3,16,1,45]

pylab.plot(x,y,tocke_x,tocke_y,'ro')

\left[\text{Line2D(\_line0)}, \text{Line2D(\_line1)}\right]

pylab.savefig('lagrange.png')

#2.Zadatak: Uočite na grafu funkcije y=3^x točke (-1,1/3), (0,1) i (1,3). Nađite polinom najmanjeg stuonja koji prolazi tim točkama i pomoću njega približno odredite 3^(1/4).

g=R.lagrange_polynomial([(-1,1/3),(0,1),(1,3)])

g

\frac{2}{3} x^{2} + \frac{4}{3} x + 1

#grafički prikaz zadane funkcije (crvena boja) i pronađenog polinoma (plava boja) sa istaknutim zadanim točkama

show(plot(lambda x: 3**x,(-1,1),rgbcolor=(1,0,0))+plot(g,-1,1)+point(((-1,1/3),(0,1),(1,3)),rgbcolor=(0,1,0),faceted=True,pointsize=30),frame=True,axes=False)

vrijednost=n(3**(1/4),digits=50);vrijednost

1.3160740129524924608192189017969990551600685902058

aproksimacija=n(g(1/4),digits=50);aproksimacija

1.3750000000000000000000000000000000000000000000000

vrijednost-aproksimacija

-0.058925987047507539180781098203000944839931409794178

reset(['x'])

#3.Zadatak: Odredite kvocijent i ostatak pri dijeljenju polinoma f(x)=x^5-3x^3-5x s polinomom g(x)=x^2-x+1.

R.<x>=QQ[]

f=x^5-3*x^3-5*x

g=x^2-x+1

#kvocijent

f//g

x^{3} + x^{2} - 3 x - 4

#ostatak

f%g

-6 x + 4

#4.Zadatak: Koliki je ostatak pri dijeljenju polinoma f(x)=x^1987-2x+5 s polinomom g(x)=x-1?

f=x^1987-2*x+5
g=x-1

#ostatak

f%g

4

#kvocijent -> dugački izraz, ne vidi se cijeli ispisan pa treba dvaput zaredom kliknuti na njega da bi se vidio čitav izraz ili se pomicati mišem udesno

f//g

x^{1986} + x^{1985} + x^{1984} + x^{1983} + x^{1982} + x^{1981} + x^{1980} + x^{1979} + x^{1978} + x^{1977} + x^{1976} + x^{1975} + x^{1974} + x^{1973} + x^{1972} + x^{1971} + x^{1970} + x^{1969} + x^{1968} + x^{1967} + x^{1966} + x^{1965} + x^{1964} + x^{1963} + x^{1962} + x^{1961} + x^{1960} + x^{1959} + x^{1958} + x^{1957} + x^{1956} + x^{1955} + x^{1954} + x^{1953} + x^{1952} + x^{1951} + x^{1950} + x^{1949} + x^{1948} + x^{1947} + x^{1946} + x^{1945} + x^{1944} + x^{1943} + x^{1942} + x^{1941} + x^{1940} + x^{1939} + x^{1938} + x^{1937} + x^{1936} + x^{1935} + x^{1934} + x^{1933} + x^{1932} + x^{1931} + x^{1930} + x^{1929} + x^{1928} + x^{1927} + x^{1926} + x^{1925} + x^{1924} + x^{1923} + x^{1922} + x^{1921} + x^{1920} + x^{1919} + x^{1918} + x^{1917} + x^{1916} + x^{1915} + x^{1914} + x^{1913} + x^{1912} + x^{1911} + x^{1910} + x^{1909} + x^{1908} + x^{1907} + x^{1906} + x^{1905} + x^{1904} + x^{1903} + x^{1902} + x^{1901} + x^{1900} + x^{1899} + x^{1898} + x^{1897} + x^{1896} + x^{1895} + x^{1894} + x^{1893} + x^{1892} + x^{1891} + x^{1890} + x^{1889} + x^{1888} + x^{1887} + x^{1886} + x^{1885} + x^{1884} + x^{1883} + x^{1882} + x^{1881} + x^{1880} + x^{1879} + x^{1878} + x^{1877} + x^{1876} + x^{1875} + x^{1874} + x^{1873} + x^{1872} + x^{1871} + x^{1870} + x^{1869} + x^{1868} + x^{1867} + x^{1866} + x^{1865} + x^{1864} + x^{1863} + x^{1862} + x^{1861} + x^{1860} + x^{1859} + x^{1858} + x^{1857} + x^{1856} + x^{1855} + x^{1854} + x^{1853} + x^{1852} + x^{1851} + x^{1850} + x^{1849} + x^{1848} + x^{1847} + x^{1846} + x^{1845} + x^{1844} + x^{1843} + x^{1842} + x^{1841} + x^{1840} + x^{1839} + x^{1838} + x^{1837} + x^{1836} + x^{1835} + x^{1834} + x^{1833} + x^{1832} + x^{1831} + x^{1830} + x^{1829} + x^{1828} + x^{1827} + x^{1826} + x^{1825} + x^{1824} + x^{1823} + x^{1822} + x^{1821} + x^{1820} + x^{1819} + x^{1818} + x^{1817} + x^{1816} + x^{1815} + x^{1814} + x^{1813} + x^{1812} + x^{1811} + x^{1810} + x^{1809} + x^{1808} + x^{1807} + x^{1806} + x^{1805} + x^{1804} + x^{1803} + x^{1802} + x^{1801} + x^{1800} + x^{1799} + x^{1798} + x^{1797} + x^{1796} + x^{1795} + x^{1794} + x^{1793} + x^{1792} + x^{1791} + x^{1790} + x^{1789} + x^{1788} + x^{1787} + x^{1786} + x^{1785} + x^{1784} + x^{1783} + x^{1782} + x^{1781} + x^{1780} + x^{1779} + x^{1778} + x^{1777} + x^{1776} + x^{1775} + x^{1774} + x^{1773} + x^{1772} + x^{1771} + x^{1770} + x^{1769} + x^{1768} + x^{1767} + x^{1766} + x^{1765} + x^{1764} + x^{1763} + x^{1762} + x^{1761} + x^{1760} + x^{1759} + x^{1758} + x^{1757} + x^{1756} + x^{1755} + x^{1754} + x^{1753} + x^{1752} + x^{1751} + x^{1750} + x^{1749} + x^{1748} + x^{1747} + x^{1746} + x^{1745} + x^{1744} + x^{1743} + x^{1742} + x^{1741} + x^{1740} + x^{1739} + x^{1738} + x^{1737} + x^{1736} + x^{1735} + x^{1734} + x^{1733} + x^{1732} + x^{1731} + x^{1730} + x^{1729} + x^{1728} + x^{1727} + x^{1726} + x^{1725} + x^{1724} + x^{1723} + x^{1722} + x^{1721} + x^{1720} + x^{1719} + x^{1718} + x^{1717} + x^{1716} + x^{1715} + x^{1714} + x^{1713} + x^{1712} + x^{1711} + x^{1710} + x^{1709} + x^{1708} + x^{1707} + x^{1706} + x^{1705} + x^{1704} + x^{1703} + x^{1702} + x^{1701} + x^{1700} + x^{1699} + x^{1698} + x^{1697} + x^{1696} + x^{1695} + x^{1694} + x^{1693} + x^{1692} + x^{1691} + x^{1690} + x^{1689} + x^{1688} + x^{1687} + x^{1686} + x^{1685} + x^{1684} + x^{1683} + x^{1682} + x^{1681} + x^{1680} + x^{1679} + x^{1678} + x^{1677} + x^{1676} + x^{1675} + x^{1674} + x^{1673} + x^{1672} + x^{1671} + x^{1670} + x^{1669} + x^{1668} + x^{1667} + x^{1666} + x^{1665} + x^{1664} + x^{1663} + x^{1662} + x^{1661} + x^{1660} + x^{1659} + x^{1658} + x^{1657} + x^{1656} + x^{1655} + x^{1654} + x^{1653} + x^{1652} + x^{1651} + x^{1650} + x^{1649} + x^{1648} + x^{1647} + x^{1646} + x^{1645} + x^{1644} + x^{1643} + x^{1642} + x^{1641} + x^{1640} + x^{1639} + x^{1638} + x^{1637} + x^{1636} + x^{1635} + x^{1634} + x^{1633} + x^{1632} + x^{1631} + x^{1630} + x^{1629} + x^{1628} + x^{1627} + x^{1626} + x^{1625} + x^{1624} + x^{1623} + x^{1622} + x^{1621} + x^{1620} + x^{1619} + x^{1618} + x^{1617} + x^{1616} + x^{1615} + x^{1614} + x^{1613} + x^{1612} + x^{1611} + x^{1610} + x^{1609} + x^{1608} + x^{1607} + x^{1606} + x^{1605} + x^{1604} + x^{1603} + x^{1602} + x^{1601} + x^{1600} + x^{1599} + x^{1598} + x^{1597} + x^{1596} + x^{1595} + x^{1594} + x^{1593} + x^{1592} + x^{1591} + x^{1590} + x^{1589} + x^{1588} + x^{1587} + x^{1586} + x^{1585} + x^{1584} + x^{1583} + x^{1582} + x^{1581} + x^{1580} + x^{1579} + x^{1578} + x^{1577} + x^{1576} + x^{1575} + x^{1574} + x^{1573} + x^{1572} + x^{1571} + x^{1570} + x^{1569} + x^{1568} + x^{1567} + x^{1566} + x^{1565} + x^{1564} + x^{1563} + x^{1562} + x^{1561} + x^{1560} + x^{1559} + x^{1558} + x^{1557} + x^{1556} + x^{1555} + x^{1554} + x^{1553} + x^{1552} + x^{1551} + x^{1550} + x^{1549} + x^{1548} + x^{1547} + x^{1546} + x^{1545} + x^{1544} + x^{1543} + x^{1542} + x^{1541} + x^{1540} + x^{1539} + x^{1538} + x^{1537} + x^{1536} + x^{1535} + x^{1534} + x^{1533} + x^{1532} + x^{1531} + x^{1530} + x^{1529} + x^{1528} + x^{1527} + x^{1526} + x^{1525} + x^{1524} + x^{1523} + x^{1522} + x^{1521} + x^{1520} + x^{1519} + x^{1518} + x^{1517} + x^{1516} + x^{1515} + x^{1514} + x^{1513} + x^{1512} + x^{1511} + x^{1510} + x^{1509} + x^{1508} + x^{1507} + x^{1506} + x^{1505} + x^{1504} + x^{1503} + x^{1502} + x^{1501} + x^{1500} + x^{1499} + x^{1498} + x^{1497} + x^{1496} + x^{1495} + x^{1494} + x^{1493} + x^{1492} + x^{1491} + x^{1490} + x^{1489} + x^{1488} + x^{1487} + x^{1486} + x^{1485} + x^{1484} + x^{1483} + x^{1482} + x^{1481} + x^{1480} + x^{1479} + x^{1478} + x^{1477} + x^{1476} + x^{1475} + x^{1474} + x^{1473} + x^{1472} + x^{1471} + x^{1470} + x^{1469} + x^{1468} + x^{1467} + x^{1466} + x^{1465} + x^{1464} + x^{1463} + x^{1462} + x^{1461} + x^{1460} + x^{1459} + x^{1458} + x^{1457} + x^{1456} + x^{1455} + x^{1454} + x^{1453} + x^{1452} + x^{1451} + x^{1450} + x^{1449} + x^{1448} + x^{1447} + x^{1446} + x^{1445} + x^{1444} + x^{1443} + x^{1442} + x^{1441} + x^{1440} + x^{1439} + x^{1438} + x^{1437} + x^{1436} + x^{1435} + x^{1434} + x^{1433} + x^{1432} + x^{1431} + x^{1430} + x^{1429} + x^{1428} + x^{1427} + x^{1426} + x^{1425} + x^{1424} + x^{1423} + x^{1422} + x^{1421} + x^{1420} + x^{1419} + x^{1418} + x^{1417} + x^{1416} + x^{1415} + x^{1414} + x^{1413} + x^{1412} + x^{1411} + x^{1410} + x^{1409} + x^{1408} + x^{1407} + x^{1406} + x^{1405} + x^{1404} + x^{1403} + x^{1402} + x^{1401} + x^{1400} + x^{1399} + x^{1398} + x^{1397} + x^{1396} + x^{1395} + x^{1394} + x^{1393} + x^{1392} + x^{1391} + x^{1390} + x^{1389} + x^{1388} + x^{1387} + x^{1386} + x^{1385} + x^{1384} + x^{1383} + x^{1382} + x^{1381} + x^{1380} + x^{1379} + x^{1378} + x^{1377} + x^{1376} + x^{1375} + x^{1374} + x^{1373} + x^{1372} + x^{1371} + x^{1370} + x^{1369} + x^{1368} + x^{1367} + x^{1366} + x^{1365} + x^{1364} + x^{1363} + x^{1362} + x^{1361} + x^{1360} + x^{1359} + x^{1358} + x^{1357} + x^{1356} + x^{1355} + x^{1354} + x^{1353} + x^{1352} + x^{1351} + x^{1350} + x^{1349} + x^{1348} + x^{1347} + x^{1346} + x^{1345} + x^{1344} + x^{1343} + x^{1342} + x^{1341} + x^{1340} + x^{1339} + x^{1338} + x^{1337} + x^{1336} + x^{1335} + x^{1334} + x^{1333} + x^{1332} + x^{1331} + x^{1330} + x^{1329} + x^{1328} + x^{1327} + x^{1326} + x^{1325} + x^{1324} + x^{1323} + x^{1322} + x^{1321} + x^{1320} + x^{1319} + x^{1318} + x^{1317} + x^{1316} + x^{1315} + x^{1314} + x^{1313} + x^{1312} + x^{1311} + x^{1310} + x^{1309} + x^{1308} + x^{1307} + x^{1306} + x^{1305} + x^{1304} + x^{1303} + x^{1302} + x^{1301} + x^{1300} + x^{1299} + x^{1298} + x^{1297} + x^{1296} + x^{1295} + x^{1294} + x^{1293} + x^{1292} + x^{1291} + x^{1290} + x^{1289} + x^{1288} + x^{1287} + x^{1286} + x^{1285} + x^{1284} + x^{1283} + x^{1282} + x^{1281} + x^{1280} + x^{1279} + x^{1278} + x^{1277} + x^{1276} + x^{1275} + x^{1274} + x^{1273} + x^{1272} + x^{1271} + x^{1270} + x^{1269} + x^{1268} + x^{1267} + x^{1266} + x^{1265} + x^{1264} + x^{1263} + x^{1262} + x^{1261} + x^{1260} + x^{1259} + x^{1258} + x^{1257} + x^{1256} + x^{1255} + x^{1254} + x^{1253} + x^{1252} + x^{1251} + x^{1250} + x^{1249} + x^{1248} + x^{1247} + x^{1246} + x^{1245} + x^{1244} + x^{1243} + x^{1242} + x^{1241} + x^{1240} + x^{1239} + x^{1238} + x^{1237} + x^{1236} + x^{1235} + x^{1234} + x^{1233} + x^{1232} + x^{1231} + x^{1230} + x^{1229} + x^{1228} + x^{1227} + x^{1226} + x^{1225} + x^{1224} + x^{1223} + x^{1222} + x^{1221} + x^{1220} + x^{1219} + x^{1218} + x^{1217} + x^{1216} + x^{1215} + x^{1214} + x^{1213} + x^{1212} + x^{1211} + x^{1210} + x^{1209} + x^{1208} + x^{1207} + x^{1206} + x^{1205} + x^{1204} + x^{1203} + x^{1202} + x^{1201} + x^{1200} + x^{1199} + x^{1198} + x^{1197} + x^{1196} + x^{1195} + x^{1194} + x^{1193} + x^{1192} + x^{1191} + x^{1190} + x^{1189} + x^{1188} + x^{1187} + x^{1186} + x^{1185} + x^{1184} + x^{1183} + x^{1182} + x^{1181} + x^{1180} + x^{1179} + x^{1178} + x^{1177} + x^{1176} + x^{1175} + x^{1174} + x^{1173} + x^{1172} + x^{1171} + x^{1170} + x^{1169} + x^{1168} + x^{1167} + x^{1166} + x^{1165} + x^{1164} + x^{1163} + x^{1162} + x^{1161} + x^{1160} + x^{1159} + x^{1158} + x^{1157} + x^{1156} + x^{1155} + x^{1154} + x^{1153} + x^{1152} + x^{1151} + x^{1150} + x^{1149} + x^{1148} + x^{1147} + x^{1146} + x^{1145} + x^{1144} + x^{1143} + x^{1142} + x^{1141} + x^{1140} + x^{1139} + x^{1138} + x^{1137} + x^{1136} + x^{1135} + x^{1134} + x^{1133} + x^{1132} + x^{1131} + x^{1130} + x^{1129} + x^{1128} + x^{1127} + x^{1126} + x^{1125} + x^{1124} + x^{1123} + x^{1122} + x^{1121} + x^{1120} + x^{1119} + x^{1118} + x^{1117} + x^{1116} + x^{1115} + x^{1114} + x^{1113} + x^{1112} + x^{1111} + x^{1110} + x^{1109} + x^{1108} + x^{1107} + x^{1106} + x^{1105} + x^{1104} + x^{1103} + x^{1102} + x^{1101} + x^{1100} + x^{1099} + x^{1098} + x^{1097} + x^{1096} + x^{1095} + x^{1094} + x^{1093} + x^{1092} + x^{1091} + x^{1090} + x^{1089} + x^{1088} + x^{1087} + x^{1086} + x^{1085} + x^{1084} + x^{1083} + x^{1082} + x^{1081} + x^{1080} + x^{1079} + x^{1078} + x^{1077} + x^{1076} + x^{1075} + x^{1074} + x^{1073} + x^{1072} + x^{1071} + x^{1070} + x^{1069} + x^{1068} + x^{1067} + x^{1066} + x^{1065} + x^{1064} + x^{1063} + x^{1062} + x^{1061} + x^{1060} + x^{1059} + x^{1058} + x^{1057} + x^{1056} + x^{1055} + x^{1054} + x^{1053} + x^{1052} + x^{1051} + x^{1050} + x^{1049} + x^{1048} + x^{1047} + x^{1046} + x^{1045} + x^{1044} + x^{1043} + x^{1042} + x^{1041} + x^{1040} + x^{1039} + x^{1038} + x^{1037} + x^{1036} + x^{1035} + x^{1034} + x^{1033} + x^{1032} + x^{1031} + x^{1030} + x^{1029} + x^{1028} + x^{1027} + x^{1026} + x^{1025} + x^{1024} + x^{1023} + x^{1022} + x^{1021} + x^{1020} + x^{1019} + x^{1018} + x^{1017} + x^{1016} + x^{1015} + x^{1014} + x^{1013} + x^{1012} + x^{1011} + x^{1010} + x^{1009} + x^{1008} + x^{1007} + x^{1006} + x^{1005} + x^{1004} + x^{1003} + x^{1002} + x^{1001} + x^{1000} + x^{999} + x^{998} + x^{997} + x^{996} + x^{995} + x^{994} + x^{993} + x^{992} + x^{991} + x^{990} + x^{989} + x^{988} + x^{987} + x^{986} + x^{985} + x^{984} + x^{983} + x^{982} + x^{981} + x^{980} + x^{979} + x^{978} + x^{977} + x^{976} + x^{975} + x^{974} + x^{973} + x^{972} + x^{971} + x^{970} + x^{969} + x^{968} + x^{967} + x^{966} + x^{965} + x^{964} + x^{963} + x^{962} + x^{961} + x^{960} + x^{959} + x^{958} + x^{957} + x^{956} + x^{955} + x^{954} + x^{953} + x^{952} + x^{951} + x^{950} + x^{949} + x^{948} + x^{947} + x^{946} + x^{945} + x^{944} + x^{943} + x^{942} + x^{941} + x^{940} + x^{939} + x^{938} + x^{937} + x^{936} + x^{935} + x^{934} + x^{933} + x^{932} + x^{931} + x^{930} + x^{929} + x^{928} + x^{927} + x^{926} + x^{925} + x^{924} + x^{923} + x^{922} + x^{921} + x^{920} + x^{919} + x^{918} + x^{917} + x^{916} + x^{915} + x^{914} + x^{913} + x^{912} + x^{911} + x^{910} + x^{909} + x^{908} + x^{907} + x^{906} + x^{905} + x^{904} + x^{903} + x^{902} + x^{901} + x^{900} + x^{899} + x^{898} + x^{897} + x^{896} + x^{895} + x^{894} + x^{893} + x^{892} + x^{891} + x^{890} + x^{889} + x^{888} + x^{887} + x^{886} + x^{885} + x^{884} + x^{883} + x^{882} + x^{881} + x^{880} + x^{879} + x^{878} + x^{877} + x^{876} + x^{875} + x^{874} + x^{873} + x^{872} + x^{871} + x^{870} + x^{869} + x^{868} + x^{867} + x^{866} + x^{865} + x^{864} + x^{863} + x^{862} + x^{861} + x^{860} + x^{859} + x^{858} + x^{857} + x^{856} + x^{855} + x^{854} + x^{853} + x^{852} + x^{851} + x^{850} + x^{849} + x^{848} + x^{847} + x^{846} + x^{845} + x^{844} + x^{843} + x^{842} + x^{841} + x^{840} + x^{839} + x^{838} + x^{837} + x^{836} + x^{835} + x^{834} + x^{833} + x^{832} + x^{831} + x^{830} + x^{829} + x^{828} + x^{827} + x^{826} + x^{825} + x^{824} + x^{823} + x^{822} + x^{821} + x^{820} + x^{819} + x^{818} + x^{817} + x^{816} + x^{815} + x^{814} + x^{813} + x^{812} + x^{811} + x^{810} + x^{809} + x^{808} + x^{807} + x^{806} + x^{805} + x^{804} + x^{803} + x^{802} + x^{801} + x^{800} + x^{799} + x^{798} + x^{797} + x^{796} + x^{795} + x^{794} + x^{793} + x^{792} + x^{791} + x^{790} + x^{789} + x^{788} + x^{787} + x^{786} + x^{785} + x^{784} + x^{783} + x^{782} + x^{781} + x^{780} + x^{779} + x^{778} + x^{777} + x^{776} + x^{775} + x^{774} + x^{773} + x^{772} + x^{771} + x^{770} + x^{769} + x^{768} + x^{767} + x^{766} + x^{765} + x^{764} + x^{763} + x^{762} + x^{761} + x^{760} + x^{759} + x^{758} + x^{757} + x^{756} + x^{755} + x^{754} + x^{753} + x^{752} + x^{751} + x^{750} + x^{749} + x^{748} + x^{747} + x^{746} + x^{745} + x^{744} + x^{743} + x^{742} + x^{741} + x^{740} + x^{739} + x^{738} + x^{737} + x^{736} + x^{735} + x^{734} + x^{733} + x^{732} + x^{731} + x^{730} + x^{729} + x^{728} + x^{727} + x^{726} + x^{725} + x^{724} + x^{723} + x^{722} + x^{721} + x^{720} + x^{719} + x^{718} + x^{717} + x^{716} + x^{715} + x^{714} + x^{713} + x^{712} + x^{711} + x^{710} + x^{709} + x^{708} + x^{707} + x^{706} + x^{705} + x^{704} + x^{703} + x^{702} + x^{701} + x^{700} + x^{699} + x^{698} + x^{697} + x^{696} + x^{695} + x^{694} + x^{693} + x^{692} + x^{691} + x^{690} + x^{689} + x^{688} + x^{687} + x^{686} + x^{685} + x^{684} + x^{683} + x^{682} + x^{681} + x^{680} + x^{679} + x^{678} + x^{677} + x^{676} + x^{675} + x^{674} + x^{673} + x^{672} + x^{671} + x^{670} + x^{669} + x^{668} + x^{667} + x^{666} + x^{665} + x^{664} + x^{663} + x^{662} + x^{661} + x^{660} + x^{659} + x^{658} + x^{657} + x^{656} + x^{655} + x^{654} + x^{653} + x^{652} + x^{651} + x^{650} + x^{649} + x^{648} + x^{647} + x^{646} + x^{645} + x^{644} + x^{643} + x^{642} + x^{641} + x^{640} + x^{639} + x^{638} + x^{637} + x^{636} + x^{635} + x^{634} + x^{633} + x^{632} + x^{631} + x^{630} + x^{629} + x^{628} + x^{627} + x^{626} + x^{625} + x^{624} + x^{623} + x^{622} + x^{621} + x^{620} + x^{619} + x^{618} + x^{617} + x^{616} + x^{615} + x^{614} + x^{613} + x^{612} + x^{611} + x^{610} + x^{609} + x^{608} + x^{607} + x^{606} + x^{605} + x^{604} + x^{603} + x^{602} + x^{601} + x^{600} + x^{599} + x^{598} + x^{597} + x^{596} + x^{595} + x^{594} + x^{593} + x^{592} + x^{591} + x^{590} + x^{589} + x^{588} + x^{587} + x^{586} + x^{585} + x^{584} + x^{583} + x^{582} + x^{581} + x^{580} + x^{579} + x^{578} + x^{577} + x^{576} + x^{575} + x^{574} + x^{573} + x^{572} + x^{571} + x^{570} + x^{569} + x^{568} + x^{567} + x^{566} + x^{565} + x^{564} + x^{563} + x^{562} + x^{561} + x^{560} + x^{559} + x^{558} + x^{557} + x^{556} + x^{555} + x^{554} + x^{553} + x^{552} + x^{551} + x^{550} + x^{549} + x^{548} + x^{547} + x^{546} + x^{545} + x^{544} + x^{543} + x^{542} + x^{541} + x^{540} + x^{539} + x^{538} + x^{537} + x^{536} + x^{535} + x^{534} + x^{533} + x^{532} + x^{531} + x^{530} + x^{529} + x^{528} + x^{527} + x^{526} + x^{525} + x^{524} + x^{523} + x^{522} + x^{521} + x^{520} + x^{519} + x^{518} + x^{517} + x^{516} + x^{515} + x^{514} + x^{513} + x^{512} + x^{511} + x^{510} + x^{509} + x^{508} + x^{507} + x^{506} + x^{505} + x^{504} + x^{503} + x^{502} + x^{501} + x^{500} + x^{499} + x^{498} + x^{497} + x^{496} + x^{495} + x^{494} + x^{493} + x^{492} + x^{491} + x^{490} + x^{489} + x^{488} + x^{487} + x^{486} + x^{485} + x^{484} + x^{483} + x^{482} + x^{481} + x^{480} + x^{479} + x^{478} + x^{477} + x^{476} + x^{475} + x^{474} + x^{473} + x^{472} + x^{471} + x^{470} + x^{469} + x^{468} + x^{467} + x^{466} + x^{465} + x^{464} + x^{463} + x^{462} + x^{461} + x^{460} + x^{459} + x^{458} + x^{457} + x^{456} + x^{455} + x^{454} + x^{453} + x^{452} + x^{451} + x^{450} + x^{449} + x^{448} + x^{447} + x^{446} + x^{445} + x^{444} + x^{443} + x^{442} + x^{441} + x^{440} + x^{439} + x^{438} + x^{437} + x^{436} + x^{435} + x^{434} + x^{433} + x^{432} + x^{431} + x^{430} + x^{429} + x^{428} + x^{427} + x^{426} + x^{425} + x^{424} + x^{423} + x^{422} + x^{421} + x^{420} + x^{419} + x^{418} + x^{417} + x^{416} + x^{415} + x^{414} + x^{413} + x^{412} + x^{411} + x^{410} + x^{409} + x^{408} + x^{407} + x^{406} + x^{405} + x^{404} + x^{403} + x^{402} + x^{401} + x^{400} + x^{399} + x^{398} + x^{397} + x^{396} + x^{395} + x^{394} + x^{393} + x^{392} + x^{391} + x^{390} + x^{389} + x^{388} + x^{387} + x^{386} + x^{385} + x^{384} + x^{383} + x^{382} + x^{381} + x^{380} + x^{379} + x^{378} + x^{377} + x^{376} + x^{375} + x^{374} + x^{373} + x^{372} + x^{371} + x^{370} + x^{369} + x^{368} + x^{367} + x^{366} + x^{365} + x^{364} + x^{363} + x^{362} + x^{361} + x^{360} + x^{359} + x^{358} + x^{357} + x^{356} + x^{355} + x^{354} + x^{353} + x^{352} + x^{351} + x^{350} + x^{349} + x^{348} + x^{347} + x^{346} + x^{345} + x^{344} + x^{343} + x^{342} + x^{341} + x^{340} + x^{339} + x^{338} + x^{337} + x^{336} + x^{335} + x^{334} + x^{333} + x^{332} + x^{331} + x^{330} + x^{329} + x^{328} + x^{327} + x^{326} + x^{325} + x^{324} + x^{323} + x^{322} + x^{321} + x^{320} + x^{319} + x^{318} + x^{317} + x^{316} + x^{315} + x^{314} + x^{313} + x^{312} + x^{311} + x^{310} + x^{309} + x^{308} + x^{307} + x^{306} + x^{305} + x^{304} + x^{303} + x^{302} + x^{301} + x^{300} + x^{299} + x^{298} + x^{297} + x^{296} + x^{295} + x^{294} + x^{293} + x^{292} + x^{291} + x^{290} + x^{289} + x^{288} + x^{287} + x^{286} + x^{285} + x^{284} + x^{283} + x^{282} + x^{281} + x^{280} + x^{279} + x^{278} + x^{277} + x^{276} + x^{275} + x^{274} + x^{273} + x^{272} + x^{271} + x^{270} + x^{269} + x^{268} + x^{267} + x^{266} + x^{265} + x^{264} + x^{263} + x^{262} + x^{261} + x^{260} + x^{259} + x^{258} + x^{257} + x^{256} + x^{255} + x^{254} + x^{253} + x^{252} + x^{251} + x^{250} + x^{249} + x^{248} + x^{247} + x^{246} + x^{245} + x^{244} + x^{243} + x^{242} + x^{241} + x^{240} + x^{239} + x^{238} + x^{237} + x^{236} + x^{235} + x^{234} + x^{233} + x^{232} + x^{231} + x^{230} + x^{229} + x^{228} + x^{227} + x^{226} + x^{225} + x^{224} + x^{223} + x^{222} + x^{221} + x^{220} + x^{219} + x^{218} + x^{217} + x^{216} + x^{215} + x^{214} + x^{213} + x^{212} + x^{211} + x^{210} + x^{209} + x^{208} + x^{207} + x^{206} + x^{205} + x^{204} + x^{203} + x^{202} + x^{201} + x^{200} + x^{199} + x^{198} + x^{197} + x^{196} + x^{195} + x^{194} + x^{193} + x^{192} + x^{191} + x^{190} + x^{189} + x^{188} + x^{187} + x^{186} + x^{185} + x^{184} + x^{183} + x^{182} + x^{181} + x^{180} + x^{179} + x^{178} + x^{177} + x^{176} + x^{175} + x^{174} + x^{173} + x^{172} + x^{171} + x^{170} + x^{169} + x^{168} + x^{167} + x^{166} + x^{165} + x^{164} + x^{163} + x^{162} + x^{161} + x^{160} + x^{159} + x^{158} + x^{157} + x^{156} + x^{155} + x^{154} + x^{153} + x^{152} + x^{151} + x^{150} + x^{149} + x^{148} + x^{147} + x^{146} + x^{145} + x^{144} + x^{143} + x^{142} + x^{141} + x^{140} + x^{139} + x^{138} + x^{137} + x^{136} + x^{135} + x^{134} + x^{133} + x^{132} + x^{131} + x^{130} + x^{129} + x^{128} + x^{127} + x^{126} + x^{125} + x^{124} + x^{123} + x^{122} + x^{121} + x^{120} + x^{119} + x^{118} + x^{117} + x^{116} + x^{115} + x^{114} + x^{113} + x^{112} + x^{111} + x^{110} + x^{109} + x^{108} + x^{107} + x^{106} + x^{105} + x^{104} + x^{103} + x^{102} + x^{101} + x^{100} + x^{99} + x^{98} + x^{97} + x^{96} + x^{95} + x^{94} + x^{93} + x^{92} + x^{91} + x^{90} + x^{89} + x^{88} + x^{87} + x^{86} + x^{85} + x^{84} + x^{83} + x^{82} + x^{81} + x^{80} + x^{79} + x^{78} + x^{77} + x^{76} + x^{75} + x^{74} + x^{73} + x^{72} + x^{71} + x^{70} + x^{69} + x^{68} + x^{67} + x^{66} + x^{65} + x^{64} + x^{63} + x^{62} + x^{61} + x^{60} + x^{59} + x^{58} + x^{57} + x^{56} + x^{55} + x^{54} + x^{53} + x^{52} + x^{51} + x^{50} + x^{49} + x^{48} + x^{47} + x^{46} + x^{45} + x^{44} + x^{43} + x^{42} + x^{41} + x^{40} + x^{39} + x^{38} + x^{37} + x^{36} + x^{35} + x^{34} + x^{33} + x^{32} + x^{31} + x^{30} + x^{29} + x^{28} + x^{27} + x^{26} + x^{25} + x^{24} + x^{23} + x^{22} + x^{21} + x^{20} + x^{19} + x^{18} + x^{17} + x^{16} + x^{15} + x^{14} + x^{13} + x^{12} + x^{11} + x^{10} + x^{9} + x^{8} + x^{7} + x^{6} + x^{5} + x^{4} + x^{3} + x^{2} + x - 1

#5.Zadatak: Odredite ostatak pri dijeljenju polinoma f(x)=x^77+x^55+x^33+x^11+1 s polinomom g(x)=x^2-1.

f=x^77+x^55+x^33+x^11+1
g=x^2-1

#ostatak

f%g

4 x + 1

#kvocijent -> dugački izraz, ne vidi se cijeli ispisan pa treba dvaput zaredom kliknuti na njega da bi se vidio čitav izraz ili se pomicati mišem udesno

f//g

x^{75} + x^{73} + x^{71} + x^{69} + x^{67} + x^{65} + x^{63} + x^{61} + x^{59} + x^{57} + x^{55} + 2 x^{53} + 2 x^{51} + 2 x^{49} + 2 x^{47} + 2 x^{45} + 2 x^{43} + 2 x^{41} + 2 x^{39} + 2 x^{37} + 2 x^{35} + 2 x^{33} + 3 x^{31} + 3 x^{29} + 3 x^{27} + 3 x^{25} + 3 x^{23} + 3 x^{21} + 3 x^{19} + 3 x^{17} + 3 x^{15} + 3 x^{13} + 3 x^{11} + 4 x^{9} + 4 x^{7} + 4 x^{5} + 4 x^{3} + 4 x

#6.Zadatak: Izračunajte vrijednost polinoma f(x)=4x^4+2x^2+1 u točki 2.

f=4*x^4+2*x^2+1

f(2)

73

#7.Zadatak: Polinom P(x)=x^4-8x^3+5x^2+2x-7 razvijte po potencijama od x+2.

x=var('x')
P=x^4-8*x^3+5*x^2+2*x-7

P.taylor(x,-2,4)

89 - {146 \left( x + 2 \right)} + {77 {\left( x + 2 \right)}^{2} } - {16 {\left( x + 2 \right)}^{3} } + {\left( x + 2 \right)}^{4}

#8.Zadatak: Za polinome A(x)=x^4+6x^3+17x^2+24x+12 i B(x)=x^3-2x^2-13x-10 odredite polinome P i Q tako da vrijedi AP+BQ=M(A,B).

R.<x>=QQ[]

A=x^4+6*x^3+17*x^2+24*x+12
B=x^3-2*x^2-13*x-10

#ako nas zanima samo najveća zajednička mjera polinoma A i B

A.gcd(B)

x^{2} + 3 x + 2

gcd(A,B)

x^{2} + 3 x + 2

#ako nas zanima sve: M(A,B), P i Q -> neće općenito dati normiranu mjeru kao u ovom slučaju. Konkretno ovdje dobivamo M(A,B)=46x^2+138x+92, P(x)=1, Q(x)=-x-8. Ako bismo mjeru normirali, tada bi bilo M(A,B)=x^2+3x+2, P(x)=1/46, Q(x)=(-1/46)x-8/46.

A.xgcd(B)

\left(46 x^{2} + 138 x + 92, 1, -x - 8\right)

xgcd(A,B)

\left(46 x^{2} + 138 x + 92, 1, -x - 8\right)

#možemo sve spremiti u varijable ukoliko nam treba za daljnje korištenje

mjera,P,Q = A.xgcd(B)

mjera

46 x^{2} + 138 x + 92

P

1

Q

-x - 8

#9.Zadatak: Za polinome A(x)=4x^5-x^4-4x^3+13x^2-3x i B(x)=3x^4-x^3-3x^2+10x-3 odredite polinome P i Q tako da vrijedi AP+BQ=M(A,B).

A=4*x^5-x^4-4*x^3+13*x^2-3*x
B=3*x^4-x^3-3*x^2+10*x-3

xgcd(A,B)

\left(\frac{1}{9} x^{3} - \frac{1}{9} x + \frac{1}{3}, 1, -\frac{4}{3} x - \frac{1}{9}\right)

gcd(A,B)

x^{3} - x + 3

#10.Zadatak: Dokažite da se razlomak (x^4+x^3+1)/(x^5+x+1) ne može više skratiti.

gcd(x^5+x+1,x^4+x^3+1)

1

#11.Zadatak: Riješite jednadžbu 5x^3-9x^2-x-2=0.

f=5*x^3-9*x^2-x-2

#možemo dobiti listu intervala unutar kojih se nalaze realne nultočke. Konkretno ovdje, naš polinom ima samo jednu realnu nultočku unutar intervala (3/2,11/4).

f.real_root_intervals()

\left[\left(\left(\frac{3}{2}, \frac{11}{4}\right), 1\right)\right]

#sa slike bismo mogli očitati da bi ta realna nultočka bila x=2, što i direktnim provjeravanjem zaista jest.

plot(f,-1,3)

#daje samo racionalne nultoče polinoma f jer je on definiran kao polinom s racionalnim nultočkama. x=2 je jednostruka nultočka.

f.roots()

\left[\left(2, 1\right)\right]

#daje sve realne nultočke od f. Uočavamo da f ima samo jednu realnu jednostruku nultočku.

f.roots(RR)

\left[\left(2.00000000000000, 1\right)\right]

#daje sve kompleksne nultočke

f.roots(CC)

$$\begin{array}{l}[\left(2.00000000000000, 1\right),\\ \left(-0.0999999999999999 - 0.435889894354067i, 1\right),\\ \left(-0.0999999999999999 + 0.435889894354067i, 1\right)]\end{array}$$

#Na primjer, ako promatramo polinom g(x)=x^2-2

#polinom g je definiran kao polinom s racionalnim koeficijentima (zbog prethone definicije R.<x>=QQ[])

g=x^2-2

#polinom g nema racionalnih nultočaka

g.roots()

\left[\right]

#polinom g ima dvije realne nultočke

g.roots(RR)

\left[\left(-1.41421356237310, 1\right), \left(1.41421356237309, 1\right)\right]

#možemo koristiti i naredbu solve

u=var('u')
solve(5*u^3-9*u^2-u-2,u)

\left[u = \frac{{-\sqrt{ 19 } i} - 1}{10}, u = \frac{{\sqrt{ 19 } i} - 1}{10}, u = 2\right]

#ili

solve([5*u^3-9*u^2-u-2==0],u)

\left[u = \frac{{-\sqrt{ 19 } i} - 1}{10}, u = \frac{{\sqrt{ 19 } i} - 1}{10}, u = 2\right]

#12.Zadatak: Riješite jednadžbu 2x^4+13x^3+25x^2+15x+9=0.

#pogledajmo intervale u kojima se nalaze realne nultočke

f=2*x^4+13*x^3+25*x^2+15*x+9

f.real_root_intervals()

$$\begin{array}{l}[\left(\left(-\frac{965829028745320677378189160381}{321943009581773547211778324968}, -\frac{1931658057490640389699315312795}{643886019163547094423556649936}\right), 2\right)]\end{array}$$

f.real_root_intervals()[0][0]

\left(-\frac{965829028745320677378189160381}{321943009581773547211778324968}, -\frac{1931658057490640389699315312795}{643886019163547094423556649936}\right)

#pogledajmo numerički koji je to zapravo interval

[n(i,digits=50) for i in f.real_root_intervals()[0][0]]

$$\begin{array}{l}[-3.0000000000000001110223024625055960658215989396981,\\ -2.9999999999999986122212192185433790254004733833419]\end{array}$$

#dakle, unutar gornjeg intervala imamo dvije realne nultočke, a kako je taj interval jako mali već predosjećamo da bi x=3 mogla biti dvostruka nultočka (naravno, to je samo naša slutnja, što ne znači da mora biti i istinita). Pogledajmo i graf koji će nas još više u to uvjeriti (uočite kako graf "dodiruje" u točki (-3,0) x-os, što zapravo znači da je -3 dvostruka nultočka.

plot(f,-5,1)

#potvrdimo i konačno sve naše slutnje

f.roots(CC)

$$\begin{array}{l}[\left(-3.00000000000000 - 7.71746789375603 \times 10^{-8}i, 1\right),\\ \left(-3.00000000000000 + 7.71746789375603 \times 10^{-8}i, 1\right),\\ \left(-0.250000000000000 - 0.661437827766148i, 1\right),\\ \left(-0.250000000000000 + 0.661437827766148i, 1\right)]\end{array}$$

solve(2*u^4+13*u^3+25*u^2+15*u+9,u)

\left[u = \frac{{-\sqrt{ 7 } i} - 1}{4}, u = \frac{{\sqrt{ 7 } i} - 1}{4}, u = -3\right]

#faktorizacija polinoma f

factor(f)

\left(2\right) \cdot (x + 3)^{2} \cdot (x^{2} + \frac{1}{2} x + \frac{1}{2})

#13.Zadatak: Riješite jednadžbu x^3+3x^2-9x-20=0.

f=x^3+3*x^2-9*x-20

#Ajmo se opet igrati malo. Pogledajmo prvo intervale u kojima se nalaze realne nultočke.

f.real_root_intervals()

$$\begin{array}{l}[\left(\left(-\frac{35}{8}, -\frac{63}{16}\right), 1\right),\\ \left(\left(-\frac{7}{2}, -\frac{7}{4}\right), 1\right),\\ \left(\left(0, \frac{7}{2}\right), 1\right)]\end{array}$$

#Dakle, naš polinom ima tri realne nultočke, što možemo vidjeti i sa grafa.

plot(f,-5,4)

#sve realne nultočke

f.roots(RR)

$$\begin{array}{l}[\left(-4.00000000000000, 1\right),\\ \left(-1.79128784747792, 1\right),\\ \left(2.79128784747792, 1\right)]\end{array}$$

solve(u^3+3*u^2-9*u-20,u)

\left[u = \frac{1 - \sqrt{ 21 }}{2}, u = \frac{\sqrt{ 21 } + 1}{2}, u = -4\right]

#14.Zadatak: Riješite jednadžbu x^4-4x^3+11x^2-14x+10=0.

f=x^4-4*x^3+11*x^2-14*x+10

#f nema realnih nultočaka

f.real_root_intervals()

\left[\right]

plot(f,-3,3)

f.roots(CC)

$$\begin{array}{l}[\left(1.00000000000000 - 2.00000000000000i, 1\right),\\ \left(1.00000000000000 - 1.00000000000000i, 1\right),\\ \left(1.00000000000000 + 1.00000000000000i, 1\right),\\ \left(1.00000000000000 + 2.00000000000000i, 1\right)]\end{array}$$

solve(u^4-4*u^3+11*u^2-14*u+10,u)

\left[u = 1 - {2 i}, u = {2 i} + 1, u = 1 - i, u = i + 1\right]

reset(['i'])

#15.Zadatak: Izračunajte (i-sqrt(3))^13.

expand((i-sqrt(3))^13)

{4096 i} - {4096 \sqrt{ 3 }}

#16.Zadatak: Riješite jednadžbu 8x^5-4x^4+2x^3-7x^2+5x-1=0.

f=8*x^5-4*x^4+2*x^3-7*x^2+5*x-1

#postoje 3 relane nultočke

f.real_root_intervals()

$$\begin{array}{l}[\left(\left(\frac{115292150460684881}{230584300921369856}, \frac{230584300921369937}{461168601842739712}\right), 3\right)]\end{array}$$

[n(k,digits=50) for k in f.real_root_intervals()[0][0]]

$$\begin{array}{l}[0.49999999999999979616999157272557374078211133601986,\\ 0.50000000000000017564075194265136730847498916789778]\end{array}$$

#naslućujemo da bi 1/2 mogla biti trostruka nultočka

plot(f,-1,2)

f.roots(CC)

$$\begin{array}{l}[\left(0.499998410073509, 1\right),\\ \left(-0.500000000000000 - 0.866025403784439i, 1\right),\\ \left(-0.500000000000000 + 0.866025403784439i, 1\right),\\ \left(0.500000794963245 - 1.37690083379653 \times 10^{-6}i, 1\right),\\ \left(0.500000794963245 + 1.37690083379653 \times 10^{-6}i, 1\right)]\end{array}$$

solve(8*u^5-4*u^4+2*u^3-7*u^2+5*u-1,u)

$$\begin{array}{l}[u = \frac{{-\sqrt{ 3 } i} - 1}{2},\\ u = \frac{{\sqrt{ 3 } i} - 1}{2},\\ u = \frac{1}{2}]\end{array}$$

factor(f)

\left(8\right) \cdot (x - \frac{1}{2})^{3} \cdot (x^{2} + x + 1)

#17.Zadatak: Riješite jednadžbu x^3-3x^2+3=0.

#sa naredbom solve nećemo baš nešto pametno dobiti

solve(u^3-3*u^2+3,u)

$$\begin{array}{l}[u = {\left( \frac{{\sqrt{ 3 } i}}{2} - \frac{1}{2} \right)}^{\frac{2}{3}} + {\left( \frac{{-\sqrt{ 3 } i}}{2} - \frac{1}{2} \right) {\left( \frac{{\sqrt{ 3 } i}}{2} - \frac{1}{2} \right)}^{\frac{1}{3}} } + 1,\\ u = {\left( \frac{{\sqrt{ 3 } i}}{2} - \frac{1}{2} \right)}^{\frac{4}{3}} + \frac{\frac{{-\sqrt{ 3 } i}}{2} - \frac{1}{2}}{{\left( \frac{{\sqrt{ 3 } i}}{2} - \frac{1}{2} \right)}^{\frac{1}{3}} } + 1,\\ u = {\left( \frac{{\sqrt{ 3 } i}}{2} - \frac{1}{2} \right)}^{\frac{1}{3}} + \frac{1}{{\left( \frac{{\sqrt{ 3 } i}}{2} - \frac{1}{2} \right)}^{\frac{1}{3}} } + 1]\end{array}$$

#ali preko roots naredbe dobivamo numerička rješenja

f=x^3-3*x^2+3

f.roots(CC)

$$\begin{array}{l}[\left(-0.879385241571817, 1\right),\\ \left(1.34729635533386, 1\right),\\ \left(2.53208888623795, 1\right)]\end{array}$$

#18.Zadatak: Riješite jednadžbu z^3+i=0.

#naredbe solve i roots u ovom slučaju ne daju ono što bismo mi htjeli

solve(u^3+i,u)

$$\begin{array}{l}[u = \frac{{{\sqrt{ 3 } {{-1 i}}^{\frac{1}{3}} } i} - {{-1 i}}^{\frac{1}{3}} }{2},\\ u = \frac{{{-\sqrt{ 3 } {{-1 i}}^{\frac{1}{3}} } i} - {{-1 i}}^{\frac{1}{3}} }{2},\\ u = {{-1 i}}^{\frac{1}{3}} ]\end{array}$$

R1.<c>=CC[]

f=c^3+i

f.roots()

$$\begin{array}{l}[\left(\frac{{{\sqrt{ 3 } {{-1 i}}^{\frac{1}{3}} } i} - {{-1 i}}^{\frac{1}{3}} }{2}, 1\right),\\ \left(\frac{{{-\sqrt{ 3 } {{-1 i}}^{\frac{1}{3}} } i} - {{-1 i}}^{\frac{1}{3}} }{2}, 1\right),\\ \left({{-1 i}}^{\frac{1}{3}} , 1\right)]\end{array}$$

#međutim, Sage ima naredbu complex_roots koja će obaviti posao kako treba

from sage.rings.polynomial.complex_roots import *

R.<x>=QQ[]
K.<i>=NumberField(x^2+1)

complex_roots(x^3+i)

$$\begin{array}{l}[\left(-0.866025403784439? - 0.500000000000000?i, 1\right),\\ \left(1i, 1\right),\\ \left(0.866025403784439? - 0.500000000000000?i, 1\right)]\end{array}$$

#19.Zadatak: Riješite jednadžbu (1+i)x^4-(1-i)x=0.

complex_roots((1+i)*x^4-(1-i)*x)

$$\begin{array}{l}[\left(0, 1\right),\\ \left(-0.866025403784439? - 0.500000000000000?i, 1\right),\\ \left(1i, 1\right),\\ \left(0.866025403784439? - 0.500000000000000?i, 1\right)]\end{array}$$