# Let alpha and beta be the zeros of the cubic polynomial x^3 + ax^2 + bx + c satisfying the equation alpha*beta + 1 = 0. prove that c^2 + ac + b +1 = 0

2
by agrawal

2015-04-14T19:34:07+05:30

let be the third root of given cubic :

i think u r quite good at maths
haha not really but ty :)
by the way in which class are u
whcih classes you people are in
class 10
2015-04-15T03:26:52+05:30

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given    αβ + 1 = 0   =>      αβ = - 1      -- (1)

α , β,  δ are the roots of the polynomial:  then it is equal to the product:

(x  - α ) (x - β)  (X - δ)
=  x³ -  (α+β+δ)  x²  + ( αβ + βδ + δα) x - αβδ

so  we have
αβ δ =  -  c
by (1)      =>  c = δ            --- (2)
α + β + δ =  -  a
α + β  = - (a + δ)            ---- (3)
αβ + βδ + δα  =  b
=>    -1 + βδ + δα  =  b
=>      βδ + δα = b + 1       --- (4)

Now to prove

c² + ac + b + 1

=  δ² + a δ + βδ + δα          using the above  (2)  , (4)
=  δ (δ + a + β + α)
=  δ (δ + a - a - δ )              using  (3)
=  δ ( 0 )
= 0