# How do you write a polynomial function of least degree with integral coefficients that has the given zeroes i, -5i?

Feb 16, 2016

$f \left(x\right) = {x}^{4} + 26 {x}^{2} + 25$

#### Explanation:

The most important thing to note here is that if a polynomial has rational coefficients and imaginary roots, the imaginary roots come in pairs.

Complex roots always come in pairs if the coefficients are integral. The pairs are always complex conjugates.

Thus, if $i$ is a root, $- i$ is a root; if $- 5 i$ is a root, $5 i$ is a root.

This gives us the function:

$f \left(x\right) = \left(x - i\right) \left(x + i\right) \left(x + 5 i\right) \left(x - 5 i\right)$

Note that there are two pairs that form the pattern $\left(a + b\right) \left(a - b\right) = {a}^{2} - {b}^{2}$. It is also useful to recall that ${i}^{2} = - 1$.

$f \left(x\right) = \left({x}^{2} - {i}^{2}\right) \left({x}^{2} - 25 {i}^{2}\right)$

$f \left(x\right) = \left({x}^{2} + 1\right) \left({x}^{2} + 25\right)$

Distribute:

$f \left(x\right) = {x}^{4} + 26 {x}^{2} + 25$