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.

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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