Prove that 3x^3 -7x^2 +5 =0 has no whole solution by contradiction

Assume that there is a whole number solution for this equation. Let's say x = a, where a is a whole number.

Substituting x = a into the equation, we get:

3a^3 - 7a^2 + 5 = 0

Now, let's consider the possible values of a. Since we're looking for whole number solutions, a can be positive, negative, or zero.

If a is positive, then 3a^3 is positive, 7a^2 is positive, and 5 is positive. The sum of positive numbers cannot be equal to zero, so this case is not possible.

If a is negative, then 3a^3 is negative, 7a^2 is positive, and 5 is positive. Again, the sum of negative and positive numbers cannot be equal to zero, so this case is also not possible.

If a is zero, then 3a^3 is zero, 7a^2 is zero, and 5 is positive. Once again, the sum of zero and positive numbers cannot be equal to zero.

Since none of the possible cases result in the equation being true, we can conclude that there is no whole number solution for the equation 3x^3 - 7x^2 + 5 = 0.

Therefore, we have proven by contradiction that the equation has no whole solution.