Grade 12th passAlgebra
if sum of two roots of the equation x^4+Px^3+Qx^2+Rx+S=0 equalsto the sum of the other two, then P^3+8R
if sum of two roots of the equation x^4+Px^3+Qx^2+Rx+S=0 equalsto the sum of the other two, then P^3+8R
1 Answer
Samyak Jain
x4 + Px3 + Qx2 + Rx + S = 0
Let a, b, c, d be the roots. It is given that sum of two roots is equal to sum of other two roots.
So a + b = c + d ...(1)
We know that a + b + c + d = – P 
a + b + a + b = – P
i.e. a + b = – P/2 = c + d ….......(2)
Also, ab + ac +ad + bc + bd + cd = Q ab + a(c + d) + b(c + d) + cd = Q
ab + (a+b)(c+d) + cd = Q ab + (– P/2)2 + cd = Q [From (1) & (2)]
i.e. ab + cd = Q – P2 / 4 …..........(3)
abc + abd + acd + bcd = – R ab(c + d) + cd(a + b) = – R
ab(a + b) + cd(a + b) = (a + b)(ab + cd) = – R [From (1)]
(– P/2)(Q – P2/4) = – R [From (2) & (3)]
– (PQ/2) + (P3/8) = – R – 4PQ + P3 = – 8R
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