Question

Write a polynomial function in standard form with the given roots: -4i

Answer 1

Answer:

$f(x)=x^2+16$

Step-by-step explanation:

By the conjugate rule, if -4i is a root, then so is +4i.  So we have 2 roots, thus, we have a second degree polynomial (namely, a quadratic).  If

x = -4i, then

x + 4i is a root.

If

x = 4i, then

x - 4i is a root.  

Having (x - 4i)(x + 4i) as roots, we can now FOIL them together to get a polynomial of least degree.

FOILing gives us

$x^2+4ix-4ix-16i^2$

Notice that the +4ix and the -4ix cancel each other out, leaving you with

$x^2-16i^2$

Since

$i^2=-1$

we can make the substitution:

$x^2-16(-1)$

which simplifies to

$x^2+16$

In function notation form:

$f(x)=x^2+16$