# What is the simplest real polynomial with roots 1 + i and 2 - i

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The polynomial has roots 1 + i and 2 - i. As it has real coefficients the roots are present in pairs of complex conjugate numbers. The simplest polynomial with these roots would also have 1 - i and 2 + i as its roots. This gives the simplest polynomial as:

(x - 1 - i)(x - 1 + i)(x - 2 + i)(x - 2 - i)

= ((x - 1)^2 + 1)((x - 2)^2 + 1)

= x^4 - 6x^3 + 15x^2 - 18x + 10

**The required polynomial is x^4 - 6x^3 + 15x^2 - 18x + 10**