What kind of roots does $x^3 - 30x^2 + 108x - 104 = 0$ have?

What kind of roots does $x^3 - 30x^2 + 108x - 104 = 0$ have?

Question

The equation $x^3 - 30x^2 + 108x - 104 = 0$ has

1. no real roots;

---

1. exactly one real root;

---

1. three distinct real roots;

---

1. a repeated root.

Suggestion

What does the Factor Theorem tell us? Could we try using small integers to find one factor?

Solution

Let $f(x) = x^3 - 30x^2 + 108x - 104$.

By searching for small integer solutions, we find that $f(2) = 8 - 120 + 216 - 104 = 0,$ so $(x - 2)$ is a factor of $f(x)$. Then we can factorise completely to find $f(x) = (x - 2)(x^2 - 28x + 52) = (x - 2)(x - 2)(x - 26),$ so $f(x)$ has a repeated root and the answer is (d).