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If $\alpha$, $\beta$ and $\gamma$ are the roots of $$x^3+2x^2+3x+3=0,$$ then what is the value of $$\left(\frac{\alpha}{\alpha+1}\right)^3+\left(\frac{\beta}{\beta+1}\right)^3+\left(\frac{\gamma}{\gamma+1}\right)^3?$$

This problem has an answer of $44$ but I am unable to find it. I had my progress on paper but gave it at the contest. Then, I tried using ChatGPT to find the answer and got $-5/2$ as the answer. Please can anyone help me figure this problem?

Reference: Test of mathematics at 10+2 level, isi problem book, pg-39, problem-272.

Note: I tried to substitute the values $\alpha/(\alpha+1)$, etc., in the equation to get a pattern but this didn’t work. Also, guessing any number by hit and trial to find any one solution also did not work. I was unable to proceed further after this.

Rócherz
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Sillyasker
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  • Please show your work and partial progress. Asking ChatGPT is not progress. – Greg Martin Nov 15 '23 at 06:47
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    I gave my progress. As this is a contest problem , most probably rote calculate is not the best possible way to solve the problem, as chatgpt did, that’s why I was looking for any pattern . But the approaches failed. Any hint to the problem?@GregMartin – Sillyasker Nov 15 '23 at 06:55
  • Your hit and trial approach was correct, but the roots are not 'nice'. Hence, avoiding of such approach in this problem is valid. Next, sum of roots should be easy to calculate, as well as its product. – Anton Vrdoljak Nov 15 '23 at 07:03

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