# Find a third-degree polynomial equation with rational coefficients that has the given numbers as roots. -5 and 1 - i

Question
Polynomials
Find a third-degree polynomial equation with rational coefficients that has the given numbers as roots. -5 and 1 - i

2020-12-30
if1-i is a root, so its complex conjugate, 1+i. Write factors(x-k) where k is a root for all 3 roots. Then multiply together.
$$(x-(-5))(x-(1-i))(x-(1+i))=(x+5)(x^{2}-x(1+i)-x(1-i)+1(1-i)(1+i))$$
$$=(x+5)(x^{2}-x-ix-x+ix+1ii^{2})$$
$$=(x+5)(x^{2}-2x+2)$$
$$=x^{3}-2x^{2}+2x+5x^{2}-10x+10$$
$$=x^{3}+3x^{2}-8x+10$$

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