State True/False.If the zeros of the polynomial

a
x
3

+3b
x
2

+3cx+d
are in A.P, then

2
b
3

−3abc+
a
2

d=0
.

Question

State True/False.If the zeros of the polynomial

a
    x
    3
   
   +3b
    x
    2
   
   +3cx+d
   are in A.P, then

2
    b
    3
   
   −3abc+
    a
    2
   
   d=0
  .

Accepted Answer

Given that the zeros of the polynomial

a
    x
    3
   
   +3b
    x
    2
   
   +3cx+d
   are in A.P.Standard form of a cubic polynomial is

f
    x
   =a
    x
    3
   
   +b
    x
    2
   
   +cx+d
   . Let

α,β,
   and

γ
   in terms of the first term and the common difference of the A.P be the zeroes. let

p
   be the first term and q be the common difference of an A.P and can be expressed as,

α=p−q,β=p
   and

γ=p+q.
  Sum of zeros of the cubic polynomial =

−
    b
    a

a=a,b=3b,c=3c,
   and

d=d

⇒α+β+γ

=−

3b
          a

⇒(p−q)+p+(p+q)

=−
        
         3b
        a

⇒3p

=−
        
         3b
        a

⇒p

=−
        b
        a

Since

β=p
  , then p is a zero of the given polynomial.

⇒f
    p
   =0

⇒a
    p
    3
   
   +3b
    p
    2
   
   +3cp+d=0

⇒a

−
        b
        a

3
   
   +3b

−
        b
        a

2
   
   +3c
    
     −
      b
      a

+d=0

⇒
      
       −
        b
        3

a
        2

+
      
       3
        b
        3

a
        2

−
      
       3bc
      a
     
     +d=0

⇒
      
       2
        b
        3
       
       −3abc+
        a
        2
       
       d

a
        2

=0

⇒2
    b
    3
   
   −3abc+
    a
    2
   
   d=0
   Therefore if the zeros of the polynomial

a
    x
    3
   
   +3b
    x
    2
   
   +3cx+d
   are in A.P, then

2
    b
    3
   
   −3abc+
    a
    2
   
   d=0.
  Given statement is true.Hence the correct option is 1.

State True/False.

If the zeros of the polynomial $a x 3 + 3 b x 2 + 3 c x + d$ are in A.P, then $2 b 3 − 3 a b c + a 2 d = 0$ .

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State True/False.If the zeros of the polynomial a x 3 +3b x 2 +3cx+d are in A.P, then 2 b 3 −3abc+ a 2 d=0 .

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State True/False.

If the zeros of the polynomial $a x 3 + 3 b x 2 + 3 c x + d$ are in A.P, then $2 b 3 − 3 a b c + a 2 d = 0$ .