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

##### 1 Answer

Feb 16, 2016

#### 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

This gives us the function:

#f(x)=(x-i)(x+i)(x+5i)(x-5i)#

Note that there are two pairs that form the pattern

#f(x)=(x^2-i^2)(x^2-25i^2)#

#f(x)=(x^2+1)(x^2+25)#

Distribute:

#f(x)=x^4+26x^2+25#