# 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)(x^{2 }- (3i)^{2}))

We know that i^{2 }= -1

(x - 1)(x^{2 }- (-9))

(x - 1)(x^{2 }+ 9)

x^{3} + 9x - x^{2} - 9

Therefore, the required polynomial is x^{3 }- x^{2} + 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 x^{3 }- x^{2} + 9x - 9.

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