Question

Write the equation for a polynomial whose roots are -1, 2i and -2i in standard form

Answer 1

A lot of the time, you get problems where you're given a polynomial and you are asked to find the roots by factoring, but in this case, you'll be doing the opposite. These are the roots, or zeroes, of the polynomial, meaning that x can equal any of these 3 when y is 0.

The first step is to get 0 on one side of the equation for each root by either adding or subtracting the root to both sides:
x = -1
x + 1 = 0
x = 2i
x - 2i = 0
x = -2i
x + 2i = 0

Now, we have 3 factors for the polynomial (x + 1), (x - 2i), and (x + 2i). To find the polynomial, multiply them all together using the FOIL method:
(x + 1)(x - 2i)(x + 2i)
(x + 1)(x² − 2ix + 2ix − 4i²)
(x + 1)(x² − 4i²)
x³ − 4i²x + x² − 4i²
Now that you've gotten to this point, you'll need to know a special property of the variable i. The variable i is an imaginary number and is equal to √(-1), the square root of negative one. So, if you square the variable i, you are actually squaring the square root of -1. What happens when you square a square root? It cancels out, giving you just -1.
x³ − 4(-1)x + x² − 4(-1)
x³ + x² + 4x + 4
And that's the answer. Hope that helps!