本题目来源于试卷: Polynomials，类别为 IB数学
[问答题]
The cubic polynomial eq b(mng8rh-sp 4u-4tx a.ki-y s5il( w mp6ogu4gation $ x^{3}+b x^{2}+c x+d=0$ has three roots $x_{1}, x_{2} $ and $x_{3}$ . By expanding the product $\left(x-x_{1}\right)\left(x-x_{2}\right)\left(x-x_{3}\right) $, show that
1. 1. $ b=-\left(x_{1}+x_{2}+x_{3}\right)$ ;
2. $ c=x_{1} x_{2}+x_{1} x_{3}+x_{2} x_{3} $
3. $d=-x_{1} x_{2} x_{3}$ .
It is given that b=-9 and c=45 for parts (b) and (c) below.
2. 1. In the case that the three roots $x_{1}, x_{2}$ and $ x_{3} $ form an arithmetic sequence, show that one of the roots is 3 .
2. Hence determine the value of d .
3. In another case the three roots form a geometric sequence. Determine the value of d .
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