Question
Thu March 25, 2010 By: Hemant Kapoor

Polynomials

Expert Reply
Sun March 28, 2010

Let α, β and γ be the roots,

comparing x3 +3px2 +3qx + r with, ax3 +bx2 +cx + d

α + β + γ = -b/a = -3p

αβ+ βγ + αγ = c/a = 3q

αβγ = -d/a = -r,

Now as α, β, γ are required to be in AP.

Let α = a-d,  β = a, γ = a+d

then,

α + β + γ = -3p

a-d + a + a + d  = -3p

3a = -3p,

a = -p

and

αβ+ βγ + αγ = 3q

(a-d)a + a(a+d) + (a-d)(a+d) = 3q

a2 - ad + a2 + ad + a2 - d2 = 3q

3a2 - d2 = 3q

3p2 - d2 = 3q

d2 =3p2 -  3q

αβγ = -r

(a-d)(a)(a+d) = -r

a(a2-d2) = -r

-p(p2-d2) = -r

(p2- 3p2 + 3q) = r/p

(3q - 2p2) = r/p

The required condition.

Regards,

Team,

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