Quadratic Prob- Medium difficulty

P(x) is a quadratic such that

(1) its leading coefficient is 1

(2) P(x) and P(P(P(x))) share a root.

Then prove that P(0)*P(1)=0

3 Answers

My solution -

1 . Observe that " P ( 0 ) " is also a root of " P " .

2 . If we let the other root be " c " , then we must have -

c P ( 0 ) = P ( 0 )

Implying that , " c = 0 , 1 " .

3 . Done .

My own solution:

Given P(x) and P(P(P(x))) share a root.

So must there be some t such that P(t)=0 and P(P(P(t)))=0 implying we must have P(P(0))=0 that is P(0) is a root of P(x)

Let P(x)= x^{2}+bx+P(0) we have

P(P(0))= P(0)^{2}+b(P(0))+P(0)=0 (^*)

Case 1) P(0)=0, its obvious that P(0)P(1)=0 and we are done.

Case 2)P(0)\neq 0
From (^*) we get 1+ b+ P(0)= 0= P(1)

Hence P(0)P(1)=0 and we have proved.

Ricky . Explain step 2.

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