<s>[INST] Given a root, determine $a$ and $b$. 
Given $p(x)= x^3 + ax^2 + bx -6$ with a zero at $x=1+i$. Determine the values of $a$ and $b$.

With the question given in its current form, would it be reasonable to assume that both $a$ and $b$ are real? Since, if so, one may use the fact that $x= 1-i$ is also a zero, and therefore find a quadratic factor and use that to proceed.
Or is there another way to proceed to answer this question without having to assume real coefficients?
 [/INST] Which says that $x^3+ax^2+bx-6$ is divisible by $$(x-1+i)(x-1-i)=(x-1)^2-i^2=x^2-2x+2$$ and since
$$x^3+ax^2+bx-6=x^3-2x^2+2x+(a+2)x^2+(b-2)x-6,$$ we obtain $a+2=-3$ and $b-2=6$,
which gives $$(a,b)=(-5,8).$$
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