Write the polynomial equation of least degree having the following roots:?

Question

Write the polynomial equation of least degree having the following roots:

+-4i
+-3i

please explain how you did it as well

thanks

Answer 1

Use the quadratic formula and work backwards.
x = -b +/- √(b^2 - 4ac) / 2a

If the root is +/- 4i, then b must be 0, which makes this easy:

+/- 4i = +/- √(-4ac) / 2a

Remove the +/-:
4i = √(-4ac) / 2a

Pull the -4 out of the square root:
4i = 2i√(ac) / 2a
4i = i√(ac) / a
4 = √(ac) / a
16 = ac / a^2
16 = c / a
c = 16a

So, to construct your polynomial, b is 0, and c must be 16 times a. The simplest polynomial, therefore, is a^2 + 16.

Answer 2

If ±4i are roots, then x² + sixteen is a ingredient, because of the fact (x+4i)(x-4i) = x² - sixteen i² = x² + sixteen. in addition, x² + 9 is a ingredient. hence, the polynomial might desire to be divisible by making use of (x² + 9)(x² + sixteen) If the polynomial has yet another root, that's going to might desire to be extra beneficial degree, so your equation is: x^4 + 25 x² + a hundred and forty four.