# How do you write a polynomial equation of least degree given the roots -1, 1, 3, -3?

Aug 11, 2018

${x}^{4} - 10 {x}^{2} + 9 = 0$

#### Explanation:

A polynomial in $x$ has a zero $a$ if and only if it has a factor $\left(x - a\right)$.

So a polynomial in $x$ with zeros $- 1$, $1$, $3$ and $- 3$ must be a multiple of:

$\left(x + 1\right) \left(x - 1\right) \left(x - 3\right) \left(x + 3\right) = \left({x}^{2} - 1\right) \left({x}^{2} - 9\right) = {x}^{4} - 10 {x}^{2} + 9$

So a polynomial equation of minimum degree with roots $- 1$, $1$, $3$ and $- 3$ is:

${x}^{4} - 10 {x}^{2} + 9 = 0$