Solution:
x3+21x2+p=0
It can be written as (x−a)(x−b)(x−c)=0
Here, roots are a,b,c.
Then, x3+21x2+p=(x−a)(x−b)(x−c)
⇒x3+21x2+p=(x−a)(x2−x(b+c)+bc)⇒x3+21x2+p=x3−ax2−x2(b+c)+ax(b+c)+xbc−abc⇒x3+21x2+p=x3+x2(−a−b−c)+x(ab+ac+bc)−abc
On comparing both sides,
−a−b−c=21; ab+ac+bc=0;−abc=p⇒a+b+c=−21 ...(i); ab+ac+bc=0;abc=−p
It is given that the roots are in AP.
a+c=2b ...(ii)
using (ii) in (i),
3b=−21⇒b=−7
Put this value in (ii),
a+c=−14
Also, abc=−p
⇒ac(−7)=−p [Using b=-7]
⇒ac=7p
Also, ab+ac+bc=0
⇒−7a+ac−7c=0 [Using b=-7]
⇒7(a+c)=ac⇒7(−14)=7p⇒p=−686
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