Question

If the zeros of the polynomial `f(x)=ax^3+3bx^2+3cx+d` are in A.P. then show that `2b^3-3abc+a^2d=0`

Answer 1

According to question
`alpha-d+alpha+alpha+d=(-3b)/a`
`3alpha=(-3b)/a`
`alpha=-b/a`
`alpha^2-dd+alpha^2+alphad+alpha^2-d^2=(3)/a`

`3alpha^2-d^2=(3c)/a`
`3b^2/a^2-h^2=(3c)/a`
`d^2=(3b^2)/a^2-(3c)/a=(3b^2-3ac)/a^2`
`alpha(alpha^2-d^2)=-d/a`
`-b/a(b^2/a^2-((3b^2-3ac)/a^2))=-d/a`
`b((b^2-3b^2+3ac)/a^2)=d`
`-2b^3+3ab=a^2d`
`2b^3-3abc+a^2d=0`.