Review question

# Does $(x-1)(x-2)\times \cdots \times (x-n) = k$ have a solution? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R7017

## Suggestion

Given a positive integer $n$ and a real number $k$, consider the following equation in $x$, $(x-1)(x-2)(x-3) \times \cdots \times (x-n) = k.$ Which of the following statements about this equation is true?

1. If $n=3$, then the equation has no real solution $x$ for some values of $k$.

2. If $n$ is even, then the equation has a real solution $x$ for any given value of $k$.

3. If $k \ge 0$ then the equation has (at least) one real solution $x$.

4. The equation never has a repeated solution $x$ for any given values of $k$ and $n$.

What would the graph of $y=(x-1)(x-2)(x-3) \times \cdots \times (x-n)$ look like for each $n$?