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.
+204 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
