Polynomials

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,

TopperLearning.