Question

Write a polynomial in standard form with zeroes set at 2i, -2i, 2

Answer 1

The polynomial equation with zeroes 2i, -2i, 2 is
`x^3 -2x^2 + 4x - 8 = 0`

Solution:

Given that zeros of polynomial are 2i, -2i, 2

To find: polynomial equation in standard form

zeros of polynomial are 2i, -2i, 2. So we can say,

x = 2i

x = -2i

x = 2

Or x - 2i = 0 and x + 2i = 0 and x - 2 = 0

Multiplying the above factors, we get the polynomial equation

`(x - 2i)(x + 2i)(x - 2) = 0\\` ------- eqn 1

Using a algebraic identity,

`(a - b)(a + b) = a^2 - b^2`

Thus
`(x - 2i)(x + 2i) = x^2 - (2i)^2`

We know that
`i^2 = -1`

`Thus (x - 2i)(x + 2i) = x^2 - (2i)^2 = x^2 -4(-1) = x^2 + 4`

Substitute the above value in eqn 1

`(x^2 + 4)(x - 2) = 0`

Multiply each term in first bracket with each term in second bracket

`x^3 -2x^2 + 4x - 8 = 0`

Thus the required equation of polynomial is found