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The fourth degree polynomial equation x^4 - 7x^3 + 4x^2 + 7x - 4 = 0 has four real roots, a, b, c, and d. What is the value of the sum 1/(abc) + 1/(abd) + 1/(acd) + 1/(bcd)?

 Apr 24, 2021
 #1
avatar+1267 
+1

Try using vietas :))

 

simplfify 1/(abc) + 1/(abd) + 1/(acd) + 1/(bcd)

Then plug in the numbers from vietas equations

 

=^._.^=

 Apr 24, 2021
 #2
avatar+118504 
+1

Simplifying  the  sum....we  have that.......

 

( abd)(acd)(bcd)   + ( abc)(acd)(bcd)  +  ( abc)(abd)(bcd)  +  ( abc)(abd)(acd)

_______________________________________________________________  =

                        (abc) ( abc)  ( acd) ( bcd)

 

(abcd)^2  ( a  +  b +  c  + d)                     (a + b + c + d)

_________________________     =      ____________

      (abcd)^3                                               abcd

 

By  Vieta

 

(a + b + c + d)  =   - (-7)  =  7

 

And

 

abcd  =  -4

 

So   we  have

 

  7

 __   =        -7  /  4

-4

 

 

cool cool cool

 Apr 24, 2021

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