No, Jesus could not have sinned. The reason will be explained by examining two doctrines concerning Christ: the hypostatic union and the communicatio idiomatum.
The hypostatic union is the teaching that within the person of Christ are two distinct natures: the divine and the human. In the one person we see the attributes of both divinity and humanity. For example, Jesus is worshiped (Matthew 2:2, 11; 14:33; Hebrews 1:6); yet he also worshiped the Father (John 17). He was called God (John 20:28; Hebrews 1:8); yet he was also called a man (Mark 15:39; John 19:5).
In the doctrine of the communicatio idiomatum we see what is called the "communication of the properties." What this means is that the attributes of both the divine and the human nature were ascribed to the single person. In other words, the person of Christ would say such things as, "Father glorify me with the glory I had with you before the foundation of the world" (John 17:5), that he descended from heaven (John 3:13), and he claimed omnipresence (Matthew 28:20). So we see that he claimed the attributes of divinity. It goes without saying that he also claimed the attributes of humanity when he ate, walked, talked, etc. So in the doctrine of the communicatio idiomatum we see that the attributes of both divinity and humanity are claimed by the person of Christ. And this is significant.
One of the attributes of being divine is the inability to sin. God is holy (1 Peter 1:16) and cannot sin. Since the attributes of divinity were ascribed to a man, Jesus, then necessarily he could not have sinned.
Jesus was tempted
Some respond by saying that Jesus was tempted, and this means he could have sinned. However, this is not what temptation means. In Psalm 106:13-15 it says "They quickly forgot His works; They did not wait for His counsel, 14But craved intensely in the wilderness, and tempted God in the desert. 15 So He gave them their request, but sent a wasting disease among them" (NASB). The Israelites tempted God, but it does not mean that God could have sinned. In this case, the temptation was a test because the Hebrew word in that Psalm is nasaw; and it means to test, try, prove, tempted, etc.
So, Jesus was offered a temptation in Matthew 4:1-11; but because of his divine nature, he could not have sinned.
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