How do you find a fourth degree polynomial given roots 3i and sqrt6?

Aug 28, 2017

${x}^{4} + 3 {x}^{2} - 54 = 0$

Explanation:

The simplest polynomial with these zeros is the quadratic:

$\left(x - 3 i\right) \left(x - \sqrt{6}\right) = {x}^{2} - \left(\sqrt{6} + 3 i\right) x + 3 \sqrt{6} i$

If we want integer coefficients, then $- 3 i$ and $- \sqrt{6}$ are also zeros and the simplest polynomial, which is a quartic, is:

$\left(x - 3 i\right) \left(x + 3 i\right) \left(x - \sqrt{6}\right) \left(x + \sqrt{6}\right) = \left({x}^{2} - {\left(3 i\right)}^{2}\right) \left({x}^{2} - {\left(\sqrt{6}\right)}^{2}\right)$

$\textcolor{w h i t e}{\left(x - 3 i\right) \left(x + 3 i\right) \left(x - \sqrt{6}\right) \left(x + \sqrt{6}\right)} = \left({x}^{2} + 9\right) \left({x}^{2} - 6\right)$

$\textcolor{w h i t e}{\left(x - 3 i\right) \left(x + 3 i\right) \left(x - \sqrt{6}\right) \left(x + \sqrt{6}\right)} = {x}^{4} + 3 {x}^{2} - 54$

Aug 28, 2017

See below.

Explanation:

The complex conjugate root theorem states that if a polynomial has a root $a + b i$, then it also has the root $a - b i$. There are also radical conjugate roots: if $a + \sqrt{b}$ is a root, then $a - \sqrt{b}$ must also be a root.

Thus, if $3 i$ and $\sqrt{6}$ are roots, then $- 3 i$ and $- \sqrt{6}$ must also be roots. Now that we have the four roots, we can establish the factors that go along with them.

$\left(x - 3 i\right) \left(x - \left(- 3 i\right)\right) \left(x - \sqrt{6}\right) \left(x - \left(- \sqrt{6}\right)\right)$

$\left(x - 3 i\right) \left(x + 3 i\right) \left(x - \sqrt{6}\right) \left(x + \sqrt{6}\right)$

We can now expand to find the fourth-degree polynomial. Since $\left(a + b\right) \left(a - b\right) = {a}^{2} - {b}^{2}$, we can rewrite the above expression as

$\left({x}^{2} - {\left(3 i\right)}^{2}\right) \left({x}^{2} - 6\right)$

$\left({x}^{2} - 9 {i}^{2}\right) \left({x}^{2} - 6\right)$

$\left({x}^{2} + 9\right) \left({x}^{2} - 6\right)$

Expanding this, we get

${x}^{4} - 6 {x}^{2} + 9 {x}^{2} - 54$

${x}^{4} + 3 {x}^{2} - 54$