Question

find the condition that the zeros of a cubic polynomial f(x)= x^3 - ax^2 + bx-c may be in A.P​

Answer 1

Answer:

2a³-9ab+27c=0

Step-by-step explanation:

let x,y,z are roots of cubic polynomial then,

• X=m-d
• y=m
• z=m+d

From equation sum of the roots equal to,

• X+y+z= -(-a)/1
• m-d+m+m+d= a
• 3m =a
• m=a/3 ........eq1

From equation product of the roots equal to,

• x.y.z =c
• (m+d)(m)(m-d)=c
• (m²-d²)(m)=c
• (a²/9 - d²)(a/3) = c (from eq1)
• d²=a²/9 - 3c/2a = (a³-27)/9a........eq2

from equation sum of product roots,

• XY+yz+zx= b
• (m-d)m+m(m+d)+ (m+d)(m-d) =b
• 3m²-d²=b
• 3(a²/9)-d²=b
• 3a²/9 - (a³-27)/9a= b. (from eq2)
• (3a³-a³+27c)/9a=b
• 2a³-9ab+27c=0