SOLUTION: Find all real numbers a such that the roots of the polynomial x^3 - 3x^2 + 17x + a form an arithmetic progression and are not all real.

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Question 1209796: Find all real numbers a such that the roots of the polynomial
x^3 - 3x^2 + 17x + a
form an arithmetic progression and are not all real.
Answer by greenestamps(13195) About Me  (Show Source):
You can put this solution on YOUR website!


f%28x%29=x%5E3-3x%5E2%2B17x%2Ba

By Vieta's Theorem, the sum of the roots is 3.

If the three roots form an arithmetic progression and have a sum of 3, then the roots are 1-p, 1, and 1+p for some number p.

So one of the roots is 1. Extract that root using synthetic division.

 1 | 1  -3  17   a
   |     1  -2  15
   +----------------
     1  -2  15  a+15

Since 1 is a root, the remainder (a+15) must be zero, so a is -15.

ANSWER: a = -15

Note the remaining binomial after extracting the root x=1 is x%5E2-2x%2B15; the roots of that binomial are 1%2Bi%2Asqrt%2814%29 and 1-i%2Asqrt%2814%29.

The three roots in arithmetic progression are 1-i%2Asqrt%2814%29, 1, and 1%2Bi%2Asqrt%2814%29.