If we know that f(7) = 0, we can say that:
x - 7 is a factor of the function f(x).
This is because when we substitute x = 7 into the function, we get:
f(7) = 7^3 - 7(7)^2 + 25(7) - 175 = 0
This means that (x - 7) is one of the factors of the function.
Now we can use polynomial division or synthetic division to find the other factors. If we divide the function f(x) by (x - 7), we obtain:
x^2 - 4x + 25
Therefore, we can say that:
f(x) = (x - 7)(x^2 - 4x + 25)
The quadratic term, x^2 - 4x + 25, has no real roots because the discriminant is negative. Therefore, we can write the factorization of f(x) in terms of complex numbers as:
f(x) = (x - 7)(x - 2 + 5i)(x - 2 - 5i)
Thus, the complete factorization of f(x) is:
f(x) = (x - 7)(x - 2 + 5i)(x - 2 - 5i)