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Question about online summer camp 2021 Algebra blue group problem P6

 
 
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Question about online summer camp 2021 Algebra blue group problem P6
by Yizhou Zhang - Monday, July 5, 2021, 6:46 PM
 

Let a, b, c, d be the roots of x^4 − 5x^2 + 3x + 1. Evaluate a^4 + b^4 + c^4 + d^4 .

I am having trouble doing this problem. While I could find a^2+b^2+c^2+d^2=19 using Vieta's formula, I am stuck and do not know how to get the answer the question is asking for. Can someone help?

 
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Re: Question about online summer camp 2021 Algebra blue group problem P6
by Areteem Professor - Monday, July 12, 2021, 2:26 PM
 

This is one of those problems where we can use the given equations to reduce some exponents.

Since you know that $a$ is a root of $x^4 -5x^2+3x+1$, then $a^4 - 5a^2 + 3a + 1 = 0$, that is, $a^4 = 5a^2 -3a -1$.

 
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