How do you write a polynomial equation of least degree given the roots -3, -2i, 2i?

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Answer 1 · 2017-04-18T13:28:14

The polynomial is $x^{3} + 3 x^{2} + 4 x + 8$ $x^3+3x^2+4x+8$

A polynomial equation of least degree given the roots $α, β$ $alpha, beta$ and $γ$ $gamma$ is

$(x - α) (x - β) (x - γ)$ $(x-alpha)(x-beta)(x-gamma)$

Hence, a polynomial equation of least degree given the roots $- 3, - 2 i$ $-3, -2i$ and $2 i$ $2i$ is

$(x - (- 3)) (x - (- 2 i)) (x - 2 i)$ $(x-(-3))(x-(-2i))(x-2i)$

= $(x + 3) (x + 2 i) (x - 2 i)$ $(x+3)(x+2i)(x-2i)$

= $(x + 3) (x^{2} - {(2 i)}^{2})$ $(x+3)(x^2-(2i)^2)$

= $(x + 3) (x^{2} - 4 i^{2})$ $(x+3)(x^2-4i^2)$

= $(x + 3) (x^{2} - 4 \times (- 1))$ $(x+3)(x^2-4×(-1))$

= $(x + 3) (x^{2} + 4)$ $(x+3)(x^2+4)$

= $x^{3} + 3 x^{2} + 4 x + 12$ $x^3+3x^2+4x+12$