Final answer:
James K. Polk was determined to expand the U.S. to the Pacific, aiming to annex New Mexico and California, leading to the Mexican-American War when peaceful negotiations failed.
Step-by-step explanation:
President James K. Polk's determination regarding the northern half of Mexico was characterized by an expansionist agenda that aimed to extend United States territory to the Pacific. Polk was eager to annex the Mexican territories of New Mexico and California, seeking to acquire them peacefully through negotiation but willing to resort to military force if necessary. This led to a controversial move where Polk sent U.S. troops under General Zachary Taylor into disputed territories, leading to a skirmish and a subsequent declaration of war on Mexico, under the pretext that Mexico had shed American blood on American soil, despite the contested nature of the territory.