We clear the ḥametz out of our homes in two ways: in thought and in deed, that is, spiritually and in practice. The removal in thought is done through nullification (bitul) of the ḥametz, declaring it ownerless and considered as mere dust. We do this nullification because we violate the prohibitions of bal yera’eh and bal yimatzei only with ḥametz that belongs to us and that we consider valuable. One who nullifies his ḥametz and considers it to be as dust does not violate any prohibitions on its account. Similarly, if he declares it ownerless, he commits no violation on its account.
In addition to bitul, we also eliminate our ḥametz in fact. The evening of the 14th, we search the entire house for ḥametz, and on the day of the 14th we eliminate it from our homes.
Although each method independently is sufficient to fulfill the requirements of Torah law, the Sages required that we remove the ḥametz using both methods, to be on the safe side. Thus, we annul the ḥametz verbally and clear it out of the house physically.
On one hand, the Sages did not want to rely on the nullification alone, lest some Jews not annul the ḥametz wholeheartedly and subsequently keep it in their homes to eat after Pesaḥ. Since they did not annul the ḥametz wholeheartedly, they violate bal yera’eh and bal yimatzei by keeping it at home (Rashi, Pesaḥim 2a). Moreover, the Sages were concerned lest, if ḥametz remained in the house, people might eat it by mistake. Therefore, they also required its physical removal from the home (Tosafot ad loc.).
Similarly, the Sages did not want to rely on the search alone, lest some Jews not manage to find all the ḥametz in their homes but then find it on Pesaḥ. In that event, there is a chance that they might wait briefly before burning it – because they would feel badly for a moment about losing their ḥametz – and, in that moment, they would violate the prohibitions of bal yera’eh and bal yimatzei. By nullifying the ḥametz before Pesaḥ, however, they would not violate the prohibitions, even if they hesitated a little before burning it (MB 434:6).
. According to MA 434:5, citing Tur, if one thoroughly checks his house and yet still an olive-sized piece of ḥametz remains, he violates the prohibitions of bal yera’eh and bal yimatzei. (This is why, in his view, the primary elimination of ḥametz at the Torah level is bitul.) However, according to Rambam, Rosh, and the other Rishonim, anyone who checks his house properly, even if he does not manage to find every last bit of ḥametz, does not violate any prohibitions for unintentionally having ḥametz in his house, since he did a proper and thorough inspection. Only if he finds ḥametz in his house on Pesaḥ and intentionally leaves it in his house because he wants the ḥametz, would he violate any prohibition. Moreover, according to Taz, even Tur is of this opinion, as I have written above.