Final answer:
To prove by contradiction that the equation 3x³ - 7x² + 5 = 0 has no integer roots, assume that it does have an integer root and then show that it leads to a contradiction.
Step-by-step explanation:
To prove by contradiction that the equation 3x³ - 7x² + 5 = 0 has no integer roots, assume that it does have an integer root. Let's assume that 'a' is an integer root. Therefore, substituting 'a' into the equation, we have 3a³ - 7a² + 5 = 0.
Now, let's consider the possible factors of 3a³, which must be either 1, 3, a, or 3a. For each of these cases, we can substitute the factors into the equation and conclude that 'a' cannot be an integer root.
Therefore, by contradiction, the equation 3x³ - 7x² + 5 = 0 has no integer roots.