Find a third-degree polynomial equation with rational coefficients that has the given numbers as roots. 1 and 3i

Solution:

Given the roots are 1, 3i and it is 3^rd degree polynomial

Since, it has imaginary roots, it occurs in pairs, hence the other root is -3i

Now, all the three roots are 1, 3i, -3i

Let us write them in factors form

(x - 1)(x - 3i)(x - (-3i))

(x - 1)(x - 3i)(x + 3i)

(x - 1)(x² - (3i)²))

We know that i² = -1

(x - 1)(x² - (-9))

(x - 1)(x² + 9)

x³ + 9x - x² - 9

Therefore, the required polynomial is x³ - x² + 9x - 9

Find a third-degree polynomial equation with rational coefficients that has the given numbers as roots. 1 and 3i

Summary:

A third-degree polynomial equation with rational coefficients that has the given numbers as roots. 1 and 3i is x³ - x² + 9x - 9.