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Let a, b, c, d, and e be the distinct roots of the equation x^5 + 7x^4 - 2 = 0.  Find a^5 + b^5 + c^5 + d^5 + e^5.

 Dec 8, 2021
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Correct option is 30

a,b are roots of x2−2cx−5d=0
⇒a+b=2c

 

and c,d are roots of x2−2ax−5b=0 
⇒c+d=2a
 

∴(a−c)+(b−d)=2(c−a)

⇒(b−d)=3(c−a)…(1)   
Since c is the root of x2−2ax−5b=0⇒c2−2ac−5b=0…(2)


Similarly a is root of x2−2cx−5d=0⇒a2−2ac−5d=0…(3)


On subtracting (3) from (2)
c2−a2=5(b−d)

 

⇒(c−a)(c+a)=5×3(c−a)……(from(1))
⇒c+a=15

 

∴a+b+c+d=2c+2a

 

=2(c+a)=2×15=30

 Dec 9, 2021

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