Thu March 25, 2010 By: Hemant Kapoor

Polynomials

1 Comments

7 years ago

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.


 

Kasim Razak 2 2 days ago
Good. I like this video/chapter/test/question/greeting
Kasim Razak 3 2 days ago
Good. I like this video/chapter/test/question/greeting
Top Contributors this Month
#1
3515 Comments 3663 Likes
#2
3229 Comments 3257 Likes
#3
3042 Comments 5820 Likes
#4
1675 Comments 3084 Likes