Q. 194.0( 11 Votes )

Find the condition that the zeros of the polynomial f(x) = x3 + 3px2 + 3qx + r may be in Arithmetic Progression.

Answer :

Given: f(x) = x3 + 3px2 + 3qx + r


Concept Used:


For a cubic polynomial: ax3 + bx2 + cx + d = 0


Sum of roots


The product of roots taken two at a time


The product of roots


Explanation:


Let the roots be,


α = a – d


β = a


γ = a + d


Sum of roots = (a – d) + a + (a + d) = 3a


Also,


Sum of roots


Therefore,


3a = –3p


a = –p


Since a is the zero of the polynomial, therefore f(a) = 0


f(a) = a3 + 3pa2 + 3qa + r = 0


a3 + 3pa2 + 3qa + r = 0


Substitute a = –p


–p3 + 3p3 – 3pq + r = 0


2p3 – 3pq + r = 0


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