polynomial

[thegersters]

A polynomial from the list below with zeroes of 2i, -2i, i, and -i would be

Is this the right answer?

x^ 4 - 5x^ 2 + 4

 

[ting]

Nearly, is it a typo or is there a similar choice, you need to change a sign.

x^ 4 - 5x^ 2 + 4, has real zeros.

It factors into, (x^2 - 4)(x^2 - 1)


Finding the zeros


x^2 - 4 = 0

x^2 = 4

x = +/- sqrt4

x = 2 and x = -2


x^2 - 1 = 0

x^2 = 1

x = +/- sqrt1

x = 1 and x = -1

 

[stapel]

The previous reply shows how to work backwards from a known polynomial to find the zeroes. To work forward from the known zeroes to the polynomial, recall how "solving" works.

Given, say, (x - 2)(x + 3) = 0, you solve the factors to get x = 2 and x = -3 as the zeroes. Going the other way, given x = 2 and x = -3 as zeroes, you then know that x - 2 = 0 and x + 3 = 0 were the factors you solved, so x - 2 and x + 3 were factors. Then you can multiply those factors to find a polynomial with those zeroes.

In this case, you have x = i, x = -i, x = 2i, and x = -2i as zeroes. Then you solve x - i = 0, x + i = 0, x - 2i = 0, and x + 2i = 0, so the factors were x - i, x + i, x - 2i, and x + 2i.

Multiply these together to get a polynomial with the required zeroes.

Hope that helps a bit.

Eliz.