Review question

# What kind of roots does $x^3 - 30x^2 + 108x - 104 = 0$ have? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R9006

## Solution

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

1. no real roots;

2. exactly one real root;

3. three distinct real roots;

4. a repeated root.

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).