What is the polynomial function of lowest degree with lead coefficient 1 and roots i, -2, and 2?

Solution:

The conjugate root theorem states that if the complex number a + bi is a root of a polynomial

P(x) in one variable with real coefficients, then the complex conjugate a - bi is also a root of

that polynomial.

Given, the roots are i, -2 and 2.

The polynomial is the product of its factors,

So, f(x) = (x - i) (x + i) (x - 2) (x + 2)

f(x) = (x²  - i²)(x² - 2² )

Using the algebraic identity (a² - b²) = (a + b) (a - b)

We know that i² = -1

f(x) = (x² + 1)(x² - 4)

f(x) = x⁴ - 4x² + x² - 4

f(x) = x⁴ - 3x² - 4

Therefore, the polynomial function of lowest degree is f(x) = x⁴ - 3x² - 4.

What is the polynomial function of lowest degree with lead coefficient 1 and roots i, -2, and 2?

Summary:

The polynomial function of lowest degree with lead coefficient 1 and roots i, -2, and 2 is f(x) =x⁴ - 3x² - 4.