Question

Write the polynomial in simplest form with roots 4, 5i, and -5i.

Answer 1

Take the complex roots first. You just need one root because complex roots always occur in pairs.
x = 5i
square both sides:
x^2 = (5i)^2 = -25
(x^2 + 25) = 0 will have the roots 5i and -5i
Now multiply (x^2 + 25) by (x-4) to get the polynomial.

Answer 2

since the roots are 4, 5i, and -5i.
the polynomial is of the form
(x-4)(x+5i)(x-5i)

Answer 3

Thank you!

Answer 4

You are welcome.

Answer 5

Could you possilbly do another example? I dont really understand it

Answer 6

Well, another way to do the same problem is what matricked suggested above.
If three roots, a, b and c are given, then the simplest polynomial with those three roots is: (x-a)(x-b)(x-c).

So here it is (x+5i)(x-5i)(x-4)
First multiply (x+5i)(x-5i). Use the identity (a+b)(a-b) = a^2 - b^2
So (x+5i)(x-5i) = x^2 - (5i)^2 = x^2 + 25
Then multiply (x^2+25)(x-4)

Answer 7

Like foils, thank you! is this for all polynomial things, no matter what roots are given?

Answer 8

Yes. If they given you any number of roots and ask for the simplest polynomial with those roots, then just proceed as shown.
This works because if 'a' is a root it implies (x-a) is a factor.

Answer 9

So if a, b, c, d, e are roots then (x-a), (x-b), (x-c), (x-d), (x-e) are factors and you can multiply them all and find the polynomial.
Example find the simplest polynomial with roots 2 and -3.
Answer: (x-2)(x- -3) or (x-2)(x+3). FOIL it and simplify to get x^2 + x - 6.