Question

find the condition that the zeroes of the polynomial f (x )= x cube - x square + q x - r may be in arithmetic progression.

Answer 1

Please check the attachment for your answer !

Answer 2

Answer:

Let zeroes be a-d, a, a+d

sum of zeroes = p

So a- d + a + a + d = p

3a = p

a = p/3

product of zeroes = r

( a-d)( a)( a+d) = r

a ( a^2 - d^2) = r

p/3 ( p^2/9 - d^2) = r

p ^2 /9 - d^2 = 3r/p

d^2 = p^2/9 - 3r/p

Sum of product of roots taking 2 at a time = q

a( a- D) + a( a+ d) + ( a-d)( a+ d) = q

a( a-d + a + d) + a^2 - d^2 = q

3a + a^2 - d^2 = q

3( p/3) + p^2/9 - p^2/9 + 3r/p = q

p + 3r/p = q

p^2 + 3r - pq = 0