# How do you write a polynomial equation of least degree given the roots -1, 1, 5?

Aug 11, 2018

${x}^{3} - 5 {x}^{2} - x + 5 = 0$

#### Explanation:

If $a$ is a zero of a polynomial in $x$, then $\left(x - a\right)$ is a factor and vice versa.

So a polynomial of minimum degree with zeros $- 1$, $1$ and $5$ is:

$\left(x + 1\right) \left(x - 1\right) \left(x - 5\right) = \left({x}^{2} - 1\right) \left(x - 5\right) = {x}^{3} - 5 {x}^{2} - x + 5$

and a polynomial equation of minimum degree with roots $- 1$, $1$ and $5$ is:

${x}^{3} - 5 {x}^{2} - x + 5 = 0$

Any non-zero constant multiple of this cubic equation is also a soltuion.