Question

Find a polynomial function of lowest degree with rational coefficients that has the given numbers as some of its zeros 2+i, 2

Answer 1

x^3 - 6x^2 + 13x - 10 We have been given the roots 2+i and 2, so two of the factors in the desired polynomial is (x - 2) = 0 (x - 2 -i ) = 0 Let's create a third root by negating the sign of the imaginary component to create the complex conjugate of the known complex root. (x - 2 + i) = 0 Now multiple (x - 2 - i) and (x - 2 + i) (x - 2 - i) (x - 2 + i) x^2 - 2x - xi -2x +4 +2i xi -2i - (-1) x^2 -4x +4 + 1 x^2 -4x + 5 And we've cancelled the imaginary term. Let's multiply this quadratic equation by the remaining factor. x^2 -4x + 5 x - 2 x^3 - 4x^2 + 5x -2x^2 +8x -10 = x^3 -6x^2 + 13x - 10 So the desired polynomial is x^3 - 6x^2 + 13x - 10