In the end of Parshas Ki Seitsei, the Torah commands us to remember the attack of Amalek against the Jewish people, when we were leaving Mitzrayim. This mitzvo is fulfilled by reading the portion that commands us with regard to this remembrance. Towards the end of the portion we are also commanded to wipe out the memory of Amalek - this means we must destroy any Amalek adult, child and animal. A person may find the command to destroy a whole nation difficult to understand.
Indeed this mitzvo troubled the righteous King, Shaul HaMelech when he was commanded to destroy the whole nation of Amalek. Understanding the underlying mistake in this episode will help us answer the question above. The Prophet tells us that, on Hashem's command, Shmuel HaNavi instructed Shaul to wipe out the whole nation, including the women, children, and animals. Shaul defeated the Amalekim in the subsequent battle, and did kill everyone with the exception of the Amalekite King, Agag, and a few animals. The gemara offers an explanation as to Shaul's reluctance to kill all the Amalekim. It tells us that Shaul made a kal v'chomer; he noted the mitzvo of egla arufa - this is a solemn ceremony that takes place upon the murder of a person in between two cities. It demonstrates the Torah's concern with regard to a single death, and its emphasis on the value of human life. Shaul reasoned that if a single human life had so much value, all the more so that is the case with regard to a whole nation. The gemara further tells us that in reaction to Shaul's 'merciful' reasoning, a Bas kol (Heavenly voice) came out, and said, "do not be overly righteous". A short time later, Shaul was pursuing David HaMelech as he felt David threatened Shaul's kingship. David took refuge with a group of Kohanim who lived in the city of Nov. Unaware of Shaul's enmity to David, they fed David and provided him with a sword. When Shaul heard about this, he ordered the murder of the whole city. At that time, another Bas Kol came out, saying, "do not be overly evil."
The Medrash makes a puzzling observation connecting these two incidents: "Anyone who is merciful in a situation where he should be cruel, eventually, he will be cruel in a situation where he should be merciful. The Medrash states clearly that it is inevitable that one who is inappropriately merciful will come to be cruel in an unsuitable manner. Why is this course of events so certain? My Rebbe, Rav Yitzchak Berkovits shlita explains that Shaul's underlying mistake was that he put his own natural emotions before the Torah's commands. Accordingly, in a situation where his natural sense of justice contradicted with a command to kill children, he chose his emotions ahead of detaching himself from his emotions in order to fulfill Hashem's word. However, in another situation, his emotions communicated to him a very different message; he perceived that David was a threat to his whole family, therefore he felt that anyone helping David was also a threat to his family and must be killed. Again, he placed his emotions before the Torah's instructions and ordered the ruthless murder of innocent people. Now we can understand the inexorable connection between Shaul's misplaced mercy and his inappropriate cruelty. A person who follows his emotions to the side of 'mercy' is nonetheless at the whim of his emotions and not morality as defined by the Torah. Therefore it is inevitable that on a separate occasion his emotions will pull him in a different direction and cause him to be overly cruel.
The account of Shaul’s failure to wipe out Amalek ended when Shmuel HaNavi personally struck down King Agag. The Ralbag brings out a remarkable point in this incident. Shmuel had Agag brought in front of him in chains. When Agag saw the righteous Shmuel, he exclaimed: “sar mar hamaves”. The Ralbag understands that Agag was saying that the bitterness of death had now gone away. This is because when he saw Shmuel he recognized his attributes of kindness and mercy, and thus he presumed that Shmuel would show mercy upon him. However, Shmuel quickly corrected Agag, telling him that he deserved to die, and he subsequently killed him. Shmuel was a merciful person because, in general, the Torah encourages the trait of mercy. However, on this occasion, Shmuel knew that mercy was inappropriate and, in this instance, the seemingly ‘cruel’ act of killing was the moral course of action because that was HaShem‘s will.
These incidents help us recognize that one cannot define morality according to his own subjective feelings and beliefs. When a person acts in such a way, he can begin to justify all kinds of evil actions. Indeed this is a common trend in secular society. People that do not believe in an objective morality feel free to define what constitutes ‘murder‘, for example. Thus, they judge that killing unborn fetuses or terminally ill people are valid courses of action. The Torah Jew recognizes that all human attempts to define morality are subject to terrible misuse. The only valid way of defining morality is by following the Torah's guidelines. Indeed morality, like everything else in the universe, was created and defined by HaShem. Accordingly, when a person finds it difficult to understand the moral nature of a Mitzvo in the Torah, this does not mean that the Mtizvo is immoral chas v’Shalom. Rather, it means that the person is following his own natural emotions and inclinations. The Torah encourages emotional expression, but only after a person has channeled his emotions through the prism of Torah. Then, he can shift his emotions to be in line with Torah morality.
With such an understanding, we realize that if HaShem commands us to destroy an entire nation, then that is the moral course of action. Indeed it is commendable to try to understand why the Torah makes such a command, and with some contemplation as to what Amalek represent, it is not difficult to understand. Nonetheless, the foundation is to recognize that the Torah conception of morality is the only valid one.
 It should be pointed out that the Minchas Chinuch (Mitzvo 604) points out that we cannot technically fulfill this mitzvo because we cannot identify who is really an Amaleki. This is because the Assyrian King, Senchariv moved around all the nations to the extent that their true identity was lost.
 Shmuel 1, Ch.15.
 This is a kind of logical argument, which is best translated as "all the more so".
 Literally meaning, the 'calf with the broken neck'.
 See Parshas Shoftim, 21:1-9 for the details of this mitzvo.
 Yoma, 22b. There are a number of difficulties with this gemara, foremost amongst them of how Shaul could disregard Hashem's clear instructions to wipe out the whole nation. See Ben Yehoyada, Anaf Yosef, and Rif in Iyun Yaakov for approaches. In this essay, we will follow the simple meaning of the gemara, that Shaul felt it was immoral to wipe out the whole nation of Amalek.
 This is part of a verse from Kohelel, Ch.7.
 Shmuel 1, Ch.21.
 Koheles Rabbah, 7:33, 16. Medrash Tanchuma, Metsora, 1.
 It should be noted that Shaul HaMelech was a very great tzaddik, and, the sins of great people recorded in Tanach are always magnified so that we can relate to them.
 Shmuel 1, Ch.15:32.
 One way to help understand what Amalek represents it to equate them to the Nazis yemach shemam. By doing this, one would find a command to destroy them as being far more understandable.