38:@0.036727:0.974518:0.057230:0.974518:0.057230:0.956333:0.036727:0.956333:0.000000:0.000000 Números reales y funciones:@0.073089:0.043661:0.318030:0.043661:0.318030:0.024916:0.073089:0.024916:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 4.1 Funciones polinómicas:@0.293660:0.231541:0.501568:0.231541:0.501568:0.214594:0.293660:0.214594:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Dentro de la clasificación de las funciones encontramos las funciones polinómicas, que :@0.293660:0.258903:0.931442:0.258903:0.931442:0.243190:0.293660:0.243190:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 son todas aquellas de la forma ( ), en donde: :@0.293660:0.275545:0.658970:0.275545:0.658970:0.259832:0.293660:0.259832:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 y:@0.517069:0.275545:0.524542:0.275545:0.524542:0.260109:0.517069:0.260109:0.000000 5:@0.528426:0.274990:0.543759:0.274990:0.543759:0.261274:0.528426:0.261274:0.000000 p x:@0.547625:0.275545:0.569069:0.275545:0.569069:0.260109:0.547625:0.260109:0.000000:0.000000:0.000000 p x:@0.405558:0.301118:0.427002:0.301118:0.427002:0.285682:0.405558:0.285682:0.000000:0.000000:0.000000 ( ) :@0.414743:0.301118:0.435764:0.301118:0.435764:0.285405:0.414743:0.285405:0.000000:0.000000:0.000000:0.000000 5:@0.435672:0.300563:0.451005:0.300563:0.451005:0.286847:0.435672:0.286847:0.000000 a x:@0.450913:0.301118:0.476319:0.301123:0.476319:0.285687:0.450913:0.285682:0.000000:0.000000:0.000000:0.000000 n:@0.463365:0.304295:0.468902:0.304295:0.468902:0.295296:0.463365:0.295296:0.000000 n:@0.476253:0.295059:0.481791:0.295059:0.481791:0.286060:0.476253:0.286060:0.000000 :@0.481752:0.301123:0.504982:0.301123:0.504982:0.285410:0.481752:0.285410:0.000000:0.000000:0.000000 1:@0.485710:0.300568:0.501043:0.300568:0.501043:0.286852:0.485710:0.286852:0.000000 a:@0.504908:0.301123:0.514167:0.301123:0.514167:0.285687:0.504908:0.285687:0.000000 n:@0.514085:0.304295:0.519623:0.304295:0.519623:0.295296:0.514085:0.295296:0.000000 2:@0.519580:0.303972:0.528519:0.303972:0.528519:0.295975:0.519580:0.295975:0.000000 1:@0.528465:0.304295:0.533702:0.304295:0.533702:0.295134:0.528465:0.295134:0.000000 x:@0.533661:0.301123:0.541116:0.301123:0.541116:0.285687:0.533661:0.285687:0.000000 n:@0.541049:0.295059:0.546586:0.295059:0.546586:0.286060:0.541049:0.286060:0.000000 2:@0.546543:0.294735:0.555482:0.294735:0.555482:0.286739:0.546543:0.286739:0.000000 1 :@0.555429:0.295059:0.562973:0.295059:0.562973:0.285898:0.555429:0.285898:0.000000:0.000000 1:@0.562926:0.300568:0.578260:0.300568:0.578260:0.286852:0.562926:0.286852:0.000000 :@0.578168:0.301123:0.582199:0.301123:0.582199:0.285410:0.578168:0.285410:0.000000 a:@0.582125:0.301123:0.591384:0.301123:0.591384:0.285687:0.582125:0.285687:0.000000 n:@0.591310:0.304295:0.596848:0.304295:0.596848:0.295296:0.591310:0.295296:0.000000 2:@0.596805:0.303972:0.605744:0.303972:0.605744:0.295975:0.596805:0.295975:0.000000 2:@0.605691:0.304295:0.610928:0.304295:0.610928:0.295134:0.605691:0.295134:0.000000 x:@0.610884:0.301123:0.618339:0.301123:0.618339:0.285687:0.610884:0.285687:0.000000 n:@0.618272:0.295059:0.623809:0.295059:0.623809:0.286060:0.618272:0.286060:0.000000 2:@0.623767:0.294735:0.632706:0.294735:0.632706:0.286739:0.623767:0.286739:0.000000 2 :@0.632652:0.295059:0.640196:0.295059:0.640196:0.285898:0.632652:0.285898:0.000000:0.000000 1:@0.640150:0.300568:0.655483:0.300568:0.655483:0.286852:0.640150:0.286852:0.000000 … :@0.655391:0.301123:0.681713:0.301123:0.681713:0.285410:0.655391:0.285410:0.000000:0.000000:0.000000 1 :@0.681621:0.300568:0.701464:0.300568:0.701464:0.286852:0.681621:0.286852:0.000000:0.000000 a x:@0.701372:0.301123:0.723207:0.301123:0.723207:0.285687:0.701372:0.285687:0.000000:0.000000:0.000000 2:@0.710561:0.304295:0.715798:0.304295:0.715798:0.295134:0.710561:0.295134:0.000000 2 :@0.723140:0.295059:0.732991:0.295059:0.732991:0.285898:0.723140:0.285898:0.000000:0.000000:0.000000 1:@0.732936:0.300568:0.748269:0.300568:0.748269:0.286852:0.732936:0.286852:0.000000 :@0.748177:0.301123:0.752208:0.301123:0.752208:0.285410:0.748177:0.285410:0.000000 a x:@0.752135:0.301123:0.773966:0.301123:0.773966:0.285687:0.752135:0.285687:0.000000:0.000000:0.000000 1:@0.761320:0.304295:0.766557:0.304295:0.766557:0.295134:0.761320:0.295134:0.000000 1 :@0.773899:0.295059:0.781443:0.295059:0.781443:0.285898:0.773899:0.285898:0.000000:0.000000 1:@0.781392:0.300568:0.796725:0.300568:0.796725:0.286852:0.781392:0.286852:0.000000 :@0.796633:0.301123:0.800665:0.301123:0.800665:0.285410:0.796633:0.285410:0.000000 a:@0.800591:0.301123:0.809850:0.301123:0.809850:0.285687:0.800591:0.285687:0.000000 0:@0.809776:0.304295:0.815013:0.304295:0.815013:0.295134:0.809776:0.295134:0.000000 a ; a:@0.293660:0.326698:0.324107:0.326698:0.324107:0.311263:0.293660:0.311263:0.000000:0.000000:0.000000:0.000000:0.000000 n:@0.302852:0.329870:0.308389:0.329870:0.308389:0.320872:0.302852:0.320872:0.000000 n:@0.324056:0.329870:0.329593:0.329870:0.329593:0.320872:0.324056:0.320872:0.000000 2:@0.329550:0.329547:0.338489:0.329547:0.338489:0.321551:0.329550:0.321551:0.000000 1:@0.338436:0.329870:0.343673:0.329870:0.343673:0.320710:0.338436:0.320710:0.000000 ; a:@0.343629:0.326698:0.359386:0.326698:0.359386:0.311263:0.343629:0.311263:0.000000:0.000000:0.000000 n:@0.359334:0.329870:0.364871:0.329870:0.364871:0.320872:0.359334:0.320872:0.000000 2:@0.364828:0.329547:0.373768:0.329547:0.373768:0.321551:0.364828:0.321551:0.000000 2:@0.373714:0.329870:0.378951:0.329870:0.378951:0.320710:0.373714:0.320710:0.000000 ; a ; a ; a:@0.378909:0.326698:0.458029:0.326698:0.458029:0.311263:0.378909:0.311263:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 …:@0.385425:0.326698:0.403833:0.326698:0.403833:0.310985:0.385425:0.310985:0.000000 2:@0.416187:0.329870:0.421424:0.329870:0.421424:0.320710:0.416187:0.320710:0.000000 1:@0.437081:0.329870:0.442318:0.329870:0.442318:0.320710:0.437081:0.320710:0.000000 0:@0.457977:0.329870:0.463214:0.329870:0.463214:0.320710:0.457977:0.320710:0.000000 son los coeficientes del polinomio cuyo grado es . :@0.463166:0.326698:0.840901:0.326698:0.840901:0.310985:0.463166:0.310985:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 n:@0.824243:0.326698:0.833741:0.326698:0.833741:0.311263:0.824243:0.311263:0.000000 a x:@0.293653:0.352271:0.315806:0.352275:0.315806:0.336840:0.293653:0.336836:0.000000:0.000000:0.000000 n:@0.302852:0.355447:0.308389:0.355447:0.308389:0.346448:0.302852:0.346448:0.000000 n:@0.315739:0.346211:0.321276:0.346211:0.321276:0.337212:0.315739:0.337212:0.000000 : término principal del polinomio. :@0.321238:0.352275:0.566477:0.352275:0.566477:0.336562:0.321238:0.336562:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 a:@0.293663:0.377848:0.302922:0.377848:0.302922:0.362413:0.293663:0.362413:0.000000 0 :@0.302852:0.381024:0.310396:0.381024:0.310396:0.371863:0.302852:0.371863:0.000000:0.000000 : término independiente.:@0.310345:0.377852:0.490302:0.377852:0.490302:0.362139:0.310345:0.362139:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Ejemplo :@0.293650:0.403425:0.359677:0.403425:0.359677:0.386755:0.293650:0.386755:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 1: 3:@0.359161:0.403425:0.413095:0.403425:0.413095:0.387712:0.359161:0.387712:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 y:@0.374568:0.403425:0.382042:0.403425:0.382042:0.387989:0.374568:0.387989:0.000000 5:@0.385429:0.402870:0.400762:0.402870:0.400762:0.389154:0.385429:0.389154:0.000000 x:@0.412966:0.403425:0.420421:0.403425:0.420421:0.387989:0.412966:0.387989:0.000000 4 :@0.420315:0.397363:0.427827:0.397363:0.427827:0.388203:0.420315:0.388203:0.000000:0.000000 1:@0.427513:0.402872:0.442846:0.402872:0.442846:0.389157:0.427513:0.389157:0.000000 :@0.442694:0.397363:0.445044:0.397363:0.445044:0.388203:0.442694:0.388203:0.000000 5:@0.444732:0.403427:0.453715:0.403427:0.453715:0.387714:0.444732:0.387714:0.000000 x:@0.453586:0.403427:0.461041:0.403427:0.461041:0.387992:0.453586:0.387992:0.000000 3 :@0.460927:0.397363:0.468439:0.397363:0.468439:0.388203:0.460927:0.388203:0.000000:0.000000 2 1 2:@0.468124:0.402872:0.554578:0.402872:0.554578:0.389157:0.468124:0.389157:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x:@0.487304:0.403427:0.494759:0.403427:0.494759:0.387992:0.487304:0.387992:0.000000 2 :@0.494642:0.397363:0.502154:0.397363:0.502154:0.388203:0.494642:0.388203:0.000000:0.000000 2:@0.521020:0.403427:0.530003:0.403427:0.530003:0.387714:0.521020:0.387714:0.000000 x:@0.529874:0.403427:0.537329:0.403427:0.537329:0.387992:0.529874:0.387992:0.000000 :@0.537206:0.397363:0.539557:0.397363:0.539557:0.388203:0.537206:0.388203:0.000000 9 es una función polinómica cuyo término principal :@0.554431:0.403427:0.931117:0.403427:0.931117:0.387714:0.554431:0.387714:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 es 3 ; el grado de la función lo da el exponente de la variable en el término principal.:@0.293674:0.420069:0.925549:0.420069:0.925549:0.404356:0.293674:0.404356:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x:@0.322150:0.420069:0.329604:0.420069:0.329604:0.404634:0.322150:0.404634:0.000000 4:@0.329533:0.414005:0.334769:0.414005:0.334769:0.404845:0.329533:0.404845:0.000000 x:@0.746658:0.420069:0.754113:0.420069:0.754113:0.404634:0.746658:0.404634:0.000000 Podemos observar además que 3 5 1 2 9. :@0.293655:0.445642:0.848169:0.445646:0.848169:0.429933:0.293655:0.429929:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 a:@0.526857:0.445642:0.536115:0.445642:0.536115:0.430207:0.526857:0.430207:0.000000 4:@0.536057:0.448818:0.541294:0.448818:0.541294:0.439657:0.536057:0.439657:0.000000 5:@0.545204:0.445091:0.560537:0.445091:0.560537:0.431375:0.545204:0.431375:0.000000 a:@0.585147:0.445646:0.594406:0.445646:0.594406:0.430210:0.585147:0.430210:0.000000 3:@0.594334:0.448818:0.599571:0.448818:0.599571:0.439657:0.594334:0.439657:0.000000 5:@0.603481:0.445091:0.618814:0.445091:0.618814:0.431375:0.603481:0.431375:0.000000 a:@0.643425:0.445646:0.652684:0.445646:0.652684:0.430210:0.643425:0.430210:0.000000 2:@0.652612:0.448818:0.657848:0.448818:0.657848:0.439657:0.652612:0.439657:0.000000 5 2:@0.661758:0.445091:0.696290:0.445091:0.696290:0.431375:0.661758:0.431375:0.000000:0.000000:0.000000 a:@0.716938:0.445646:0.726197:0.445646:0.726197:0.430210:0.716938:0.430210:0.000000 1:@0.726130:0.448818:0.731367:0.448818:0.731367:0.439657:0.726130:0.439657:0.000000 5:@0.735277:0.445091:0.750610:0.445091:0.750610:0.431375:0.735277:0.431375:0.000000 a:@0.775221:0.445646:0.784480:0.445646:0.784480:0.430210:0.775221:0.430210:0.000000 0:@0.784408:0.448818:0.789644:0.448818:0.789644:0.439657:0.784408:0.439657:0.000000 5 2:@0.793554:0.445091:0.828086:0.445091:0.828086:0.431375:0.793554:0.431375:0.000000:0.000000:0.000000 Ejemplo 2:@0.293668:0.471219:0.372838:0.471219:0.372838:0.454549:0.293668:0.454549:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 : identifiquemos el grado de los siguientes polinomios y elaboremos sus :@0.373041:0.471219:0.930968:0.471219:0.930968:0.455506:0.373041:0.455506:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 respectivas gráficas.:@0.293668:0.487860:0.436555:0.487860:0.436555:0.472148:0.293668:0.472148:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 1. :@0.293660:0.513440:0.309730:0.513440:0.309730:0.497727:0.293660:0.497727:0.000000:0.000000:0.000000 P x:@0.323861:0.513440:0.344809:0.513440:0.344809:0.498004:0.323861:0.498004:0.000000:0.000000:0.000000 ( ) 5 es un polinomio de grado 0. Su gráfica es una recta paralela al eje .:@0.332550:0.513440:0.872213:0.513440:0.872213:0.497727:0.332550:0.497727:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 5:@0.353479:0.512885:0.368812:0.512885:0.368812:0.499169:0.353479:0.499169:0.000000 x:@0.861629:0.513440:0.869084:0.513440:0.869084:0.498004:0.861629:0.498004:0.000000 2. :@0.293660:0.530082:0.309730:0.530082:0.309730:0.514369:0.293660:0.514369:0.000000:0.000000:0.000000 P x:@0.323861:0.530082:0.344809:0.530082:0.344809:0.514646:0.323861:0.514646:0.000000:0.000000:0.000000 ( ) 2 3 es un polinomio de grado 1. Su gráfica es una recta.:@0.332550:0.530082:0.813667:0.530082:0.813667:0.514369:0.332550:0.514369:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 5 2:@0.353479:0.529527:0.388011:0.529527:0.388011:0.515811:0.353479:0.515811:0.000000:0.000000:0.000000 x:@0.396828:0.530082:0.404283:0.530082:0.404283:0.514646:0.396828:0.514646:0.000000 1:@0.408167:0.529527:0.423500:0.529527:0.423500:0.515811:0.408167:0.515811:0.000000 3. :@0.293660:0.546723:0.309730:0.546723:0.309730:0.531011:0.293660:0.531011:0.000000:0.000000:0.000000 P x:@0.323861:0.546723:0.344809:0.546723:0.344809:0.531288:0.323861:0.531288:0.000000:0.000000:0.000000 ( ) 3 :@0.332550:0.546723:0.417394:0.546723:0.417394:0.531011:0.332550:0.531011:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 5:@0.353479:0.546169:0.368812:0.546169:0.368812:0.532453:0.353479:0.532453:0.000000 x:@0.381568:0.546723:0.389023:0.546723:0.389023:0.531288:0.381568:0.531288:0.000000 2:@0.388973:0.540659:0.394210:0.540659:0.394210:0.531499:0.388973:0.531499:0.000000 1 2:@0.398121:0.546169:0.443293:0.546169:0.443293:0.532453:0.398121:0.532453:0.000000:0.000000:0.000000 x :@0.417320:0.546723:0.446514:0.546723:0.446514:0.531288:0.417320:0.531288:0.000000:0.000000:0.000000:0.000000 1 es un polinomio de grado 2. Su gráfica es una parábola.:@0.446440:0.546723:0.860442:0.546723:0.860442:0.531011:0.446440:0.531011:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 4. :@0.293660:0.563365:0.309730:0.563365:0.309730:0.547653:0.293660:0.547653:0.000000:0.000000:0.000000 P x:@0.323861:0.563365:0.345067:0.563365:0.345067:0.547930:0.323861:0.547930:0.000000:0.000000:0.000000 ( ) 1 es un polinomio de grado 3. Su gráfica a veces es llamada parábola :@0.332678:0.563365:0.930852:0.563365:0.930852:0.547653:0.332678:0.547653:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 5:@0.354822:0.562810:0.370156:0.562810:0.370156:0.549095:0.354822:0.549095:0.000000 x:@0.374978:0.563365:0.382433:0.563365:0.382433:0.547930:0.374978:0.547930:0.000000 3:@0.382477:0.557301:0.387714:0.557301:0.387714:0.548141:0.382477:0.548141:0.000000 1:@0.392514:0.562810:0.407847:0.562810:0.407847:0.549095:0.392514:0.549095:0.000000 :@0.421708:0.563365:0.425021:0.563365:0.425021:0.547930:0.421708:0.547930:0.000000 cúbica.:@0.323763:0.580007:0.375082:0.580007:0.375082:0.564294:0.323763:0.564294:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 0:@0.472180:0.748227:0.480346:0.748227:0.480346:0.733943:0.472180:0.733943:0.000000 2:@0.457287:0.809222:0.471227:0.809222:0.471227:0.796753:0.457287:0.796753:0.000000 5:@0.471227:0.809726:0.479393:0.809726:0.479393:0.795442:0.471227:0.795442:0.000000 5:@0.570642:0.751152:0.578809:0.751152:0.578809:0.736868:0.570642:0.736868:0.000000 5:@0.469938:0.675343:0.478104:0.675343:0.478104:0.661059:0.469938:0.661059:0.000000 2:@0.382571:0.749122:0.396510:0.749122:0.396510:0.736653:0.382571:0.736653:0.000000 5:@0.396510:0.749626:0.404676:0.749626:0.404676:0.735342:0.396510:0.735342:0.000000 :@0.610856:0.748891:0.614887:0.748891:0.614887:0.733179:0.610856:0.733179:0.000000 y:@0.688768:0.720955:0.695562:0.720955:0.695562:0.706922:0.688768:0.706922:0.000000 3 :@0.695502:0.720955:0.757187:0.720955:0.757187:0.706670:0.695502:0.706670:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 5:@0.699100:0.720450:0.713039:0.720450:0.713039:0.707981:0.699100:0.707981:0.000000 x:@0.724636:0.720955:0.731413:0.720955:0.731413:0.706922:0.724636:0.706922:0.000000 2:@0.731351:0.715442:0.736112:0.715442:0.736112:0.707114:0.731351:0.707114:0.000000 1 2:@0.739666:0.720450:0.780731:0.720450:0.780731:0.707981:0.739666:0.707981:0.000000:0.000000:0.000000 x :@0.757120:0.720955:0.766842:0.720955:0.766842:0.706922:0.757120:0.706922:0.000000:0.000000 1:@0.780647:0.720955:0.788814:0.720955:0.788814:0.706670:0.780647:0.706670:0.000000 y:@0.688293:0.745968:0.695087:0.745968:0.695087:0.731936:0.688293:0.731936:0.000000 1:@0.695020:0.745968:0.756695:0.745970:0.756695:0.731686:0.695020:0.731683:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 5:@0.698618:0.745463:0.712557:0.745463:0.712557:0.732995:0.698618:0.732995:0.000000 x:@0.716071:0.745968:0.722849:0.745968:0.722849:0.731936:0.716071:0.731936:0.000000 3:@0.722758:0.748854:0.727519:0.748854:0.727519:0.740526:0.722758:0.740526:0.000000 1:@0.731075:0.745466:0.745014:0.745466:0.745014:0.732997:0.731075:0.732997:0.000000 y:@0.687584:0.669492:0.694378:0.669492:0.694378:0.655460:0.687584:0.655460:0.000000 5:@0.694317:0.669492:0.723535:0.669492:0.723535:0.655208:0.694317:0.655208:0.000000:0.000000:0.000000:0.000000 5:@0.697915:0.668988:0.711855:0.668988:0.711855:0.656519:0.697915:0.656519:0.000000 y:@0.687088:0.694505:0.693882:0.694505:0.693882:0.680473:0.687088:0.680473:0.000000 2 3:@0.693815:0.694505:0.772749:0.694505:0.772749:0.680221:0.693815:0.680221:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 5 2:@0.697413:0.694001:0.728806:0.694001:0.728806:0.681532:0.697413:0.681532:0.000000:0.000000:0.000000 x:@0.736821:0.694505:0.743599:0.694505:0.743599:0.680473:0.736821:0.680473:0.000000 1:@0.747130:0.694001:0.761069:0.694001:0.761069:0.681532:0.747130:0.681532:0.000000 Figura 4.1:@0.322121:0.829924:0.373447:0.829924:0.373447:0.818497:0.322121:0.818497:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 4.2 Gráfica de funciones polinómicas :@0.293660:0.861806:0.585704:0.861806:0.585704:0.844859:0.293660:0.844859:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 En la gráfica de una función polinómica, el coeficiente del término principal determina :@0.293660:0.889168:0.930875:0.889169:0.930875:0.873457:0.293660:0.873455:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 a:@0.680746:0.889168:0.690005:0.889168:0.690005:0.873733:0.680746:0.873733:0.000000 n:@0.689880:0.892342:0.695417:0.892342:0.695417:0.883343:0.689880:0.883343:0.000000 el comportamiento de la gráfica a la izquierda y a la derecha, tal como se muestra en los :@0.293663:0.905811:0.931407:0.905811:0.931407:0.890099:0.293663:0.890099:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 siguientes ejemplos de gráficas de algunas funciones polinómicas de grado par y de :@0.293663:0.922453:0.931399:0.922453:0.931399:0.906740:0.293663:0.906740:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 grado impar.:@0.293663:0.939095:0.385575:0.939095:0.385575:0.923382:0.293663:0.923382:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Tema 4. Funciones polinómicas y racionales. Asíntotas verticales y horizontales:@0.073089:0.092690:0.927239:0.092690:0.927239:0.069846:0.073089:0.069846:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Considera la función polinómica ( ) 5 6. :@0.307891:0.165069:0.682262:0.165069:0.682262:0.149357:0.307891:0.149357:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 f x:@0.546209:0.165069:0.563089:0.165069:0.563089:0.149634:0.546209:0.149634:0.000000:0.000000:0.000000 5:@0.571759:0.164515:0.587092:0.164515:0.587092:0.150799:0.571759:0.150799:0.000000 x:@0.590957:0.165069:0.598412:0.165069:0.598412:0.149634:0.590957:0.149634:0.000000 2:@0.598344:0.159006:0.603581:0.159006:0.603581:0.149845:0.598344:0.149845:0.000000 2:@0.607492:0.164515:0.622825:0.164515:0.622825:0.150799:0.607492:0.150799:0.000000 x:@0.635582:0.165069:0.643036:0.165069:0.643036:0.149634:0.635582:0.149634:0.000000 1:@0.646920:0.164515:0.662254:0.164515:0.662254:0.150799:0.646920:0.150799:0.000000 Encuentra las raíces del polinomio ( ).:@0.307896:0.188853:0.584409:0.188853:0.584409:0.173141:0.307896:0.173141:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 f x:@0.559596:0.188853:0.576476:0.188853:0.576476:0.173418:0.559596:0.173418:0.000000:0.000000:0.000000 Reto matemático:@0.302768:0.139600:0.437491:0.139600:0.437491:0.122653:0.302768:0.122653:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Si y = ( ) es una función :@0.082576:0.347977:0.248765:0.347977:0.248765:0.333693:0.082576:0.333693:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 P x:@0.121420:0.347977:0.140396:0.347977:0.140396:0.333945:0.121420:0.333945:0.000000:0.000000:0.000000 polinómica y , :@0.082564:0.363106:0.218995:0.363106:0.218995:0.348822:0.082564:0.348822:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 a:@0.171186:0.363106:0.179604:0.363106:0.179604:0.349074:0.171186:0.349074:0.000000 [:@0.183068:0.362980:0.197007:0.362980:0.197007:0.350208:0.183068:0.350208:0.000000 R:@0.200454:0.362324:0.212536:0.362324:0.212536:0.350599:0.200454:0.350599:0.000000 entonces es cierto que :@0.082564:0.378235:0.247543:0.378235:0.247543:0.363951:0.082564:0.363951:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 a:@0.235461:0.378235:0.243878:0.378235:0.243878:0.364203:0.235461:0.364203:0.000000 es un cero de ( ) si ( ) :@0.082564:0.393364:0.242858:0.393364:0.242858:0.379080:0.082564:0.379080:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 P x:@0.174399:0.393364:0.193376:0.393364:0.193376:0.379332:0.174399:0.379332:0.000000:0.000000:0.000000 P a:@0.214243:0.393364:0.234859:0.393364:0.234859:0.379332:0.214243:0.379332:0.000000:0.000000:0.000000 5:@0.082564:0.407988:0.096503:0.407988:0.096503:0.395520:0.082564:0.395520:0.000000 0 y que es un :@0.096386:0.408493:0.230341:0.408493:0.230341:0.394208:0.096386:0.394208:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x:@0.151692:0.408493:0.158469:0.408493:0.158469:0.394461:0.151692:0.394461:0.000000 2:@0.161933:0.407988:0.175872:0.407988:0.175872:0.395520:0.161933:0.395520:0.000000 a:@0.179319:0.408493:0.187736:0.408493:0.187736:0.394461:0.179319:0.394461:0.000000 factor de ( ).:@0.082581:0.423622:0.170718:0.423622:0.170718:0.409337:0.082581:0.409337:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 P x:@0.144596:0.423622:0.163573:0.423622:0.163573:0.409590:0.144596:0.409590:0.000000:0.000000:0.000000 Competencia :@0.078581:0.304693:0.187141:0.304693:0.187141:0.287746:0.078581:0.287746:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 matemática :@0.078581:0.321335:0.176057:0.321335:0.176057:0.304388:0.078581:0.304388:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 t.ly/Q-nrs:@0.104396:0.932345:0.171356:0.932345:0.171356:0.916633:0.104396:0.916633:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Funciones :@0.080269:0.781187:0.164352:0.781187:0.164352:0.764240:0.080269:0.764240:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 polinómicas :@0.080269:0.797828:0.184086:0.797828:0.184086:0.780882:0.080269:0.780882:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 y racionales:@0.080269:0.814470:0.173353:0.814470:0.173353:0.797523:0.080269:0.797523:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 las raíces son: 2 3:@0.618264:0.185846:0.734239:0.185846:0.734239:0.175847:0.618264:0.175847:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x:@0.683542:0.185846:0.688287:0.185846:0.688287:0.176024:0.683542:0.176024:0.000000 5:@0.690852:0.185493:0.700609:0.185493:0.700609:0.176765:0.690852:0.176765:0.000000 x:@0.708891:0.185846:0.713635:0.185846:0.713635:0.176024:0.708891:0.176024:0.000000 5:@0.716200:0.185493:0.725958:0.185493:0.725958:0.176765:0.716200:0.176765:0.000000 MUESTRA EDITORIAL:@0.248902:0.947499:0.888171:0.113285:0.743922:0.050540:0.104653:0.884754:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Muestra editorial solo para fines didácticos – Prohibida su venta:@0.095585:0.659700:0.095585:0.341447:0.077558:0.341447:0.077558:0.659700:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000