Gloucester is too easily swayed by one son against the other. He is quick to believe the worst of Edgar, based on Edmund's lies. The foundation of favoritism that has existed in their relationship exposed Gloucester's insecurities, thus making it not at all difficult for Edmund to convince him of Edgar's perfidy.
Without the trust in his son's judgment, Gloucester rejects Edgar out of hand. He is influenced by appearance and strength of personaility, rather than on past experience of Edgar's loyalty and love. This mirrors Lear's rejection of Cordelia.
Gloucester confesses in the opening scene that he is an adulterer, and that the result of his adultery is his illegitimate son Edmund. Gloucester ofcourse has a legitimate son Edgar who will legally inherit his title and his wealth.
Thus, Gloucester's dilemma is to be an impartial father to both his legitimate and illegitimate sons. Gloucester himself admits that initially he was embarrassed by Edmund but now he has got accustomed to it. Gloucester accepts Edmund as his son grudgingly and reluctantly and complains that he had to bring him up at some expense and trouble - "hath been at my charge." He considers his birth a mistake,"this knave came saucily into the/world before he was sent for." Finally. Gloucester adds insult to injury by saying that he was forced to "acknowledge" his "whoreson." No wonder Edmund decides to revenge his humiliation.
In the next scene, Edmund easily tricks his father to believing that Edgar his legitimate son and legal heir is conspiring to kill him. Gloucester certainly is hasty in judging Edgar to be an "unnatural, detested/brutish villain." It is plain that the so called written and "auricular assurance" that Emund offers to Gloucester as proof of Edgar's treachery have been contrived. But Gloucester is easily fooled by Edmund because as Edmund himself remarks scornfully, he is "a credulous father!"