Question

Find the condition that the zeros of the polynomial $f(x)=x^{3}+3px^{2} +3qx+r$ may be in A.P.

Answer 1

SOLUTION :  

Let, α = a - d, β = a and γ = a + d be the zeroes of the polynomial.

Given : f(x) = x³ + 3px² + 3qx + r

Sum of zeroes = −coefficient of x² / coefficient of x³

α + β + γ = −b/a

α + β + γ = - 3p/1 = -3p

(a – d) +( a) + (a + d) = -3p

a + a + a -d -d = -3p

3a = -3p

a = -3p × ⅓ = -p

a = -p …………………..(1)

Since, a is the zero of the polynomial f(x),

Therefore, f(a) = 0

f(a)= a³ +3pa² + 3qa + r  

a³ +3pa² + 3qa + r = 0

On Substituting a = -p ,

= (−p)³ + 3p(-p)² + 3q(-p) + r=0

= −p³ +3p³ – 3pq + r=0

= 2p³ –3pq + r=0

Hence, the condition for the Given polynomial is 2p³ –3pq + r = 0.

HOPE THIS ANSWER WILL HELP YOU….

Answer 2

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