SOLUTION: There are integers $b,c$ for which both roots of the polynomial $x^2-x-1$ are also roots of the polynomial $x^5-bx-c$. Determine the product $bc$.

Algebra ->  Square-cubic-other-roots -> SOLUTION: There are integers $b,c$ for which both roots of the polynomial $x^2-x-1$ are also roots of the polynomial $x^5-bx-c$. Determine the product $bc$.      Log On


   

Question 1075769: There are integers $b,c$ for which both roots of the polynomial $x^2-x-1$ are also roots of the polynomial $x^5-bx-c$. Determine the product $bc$.
Answer by ikleyn(52933) About Me  (Show Source):
You can put this solution on YOUR website!
.
1.  The roots of the polynomial x%5E2+-+x+-1 are

    x%5B1%2C2%5D = %281+%2B-+sqrt%281%2B4%29%29%2F2 = %281+%2B-+sqrt%285%29%29%2F2.


2.  Substitute the root (the number) %281%2Bsqrt%285%29%29%2F2 into the polynomial x%5E5+-bx+-c.

    According to the condition, you will get the equation 

    %28%281%2Bsqrt%285%29%29%2F2%29%5E5 - b%2A%281%2Bsqrt%285%29%29%2F2 - c = 0.


3.  From this point, read my solution to the problem

    https://www.algebra.com/algebra/homework/Expressions-with-variables/Expressions-with-variables.faq.question.1075766.html

    
    https://www.algebra.com/algebra/homework/Expressions-with-variables/Expressions-with-variables.faq.question.1075766.html



4.  Armed with these hints, you are ready / are prepared to complete this assignment on your own.


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