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Sunday, August 29, 2021

Find all polynomials P(x) such that: P(x-1).P(x+1)=P(x2-1)

 Find all polynomials P(x) such that: P(x-1).P(x+1)=P(x2-1)

Solution

Suppose that α is a root of P(x)  then, P(α)=0

Therefore, P((α+1)-1)=0

Following, P(x-1).P(x+1)=P(x2-1)  (). We replace the value of x=α+1

                      P((α+1)2-1)=0

We easily see that (α+1)2-1 is also a root of P(x)

We will again replace x=(α+1)2in to the equation ()

                    P((α+1)4-1)=0

We easily see that (α+1)4-1) is another root of P(x)

Inductively, (α+1)2k-1 is the root k0

So far: If α is a root of P(x) then so is (α+1)2k-1 for all k1

Therefore, α=0 ; α=-1 ; α=-2

Then, P(x)=xl(x+1)m(x+2)n which A=1

            P(x-1)=(x-1)lxm(x+1)n

            P(x+1)=(x+1)l(x+2)m(x+3)n

            P(x2-1)=(x2-1)l(x2)m(x2+1)n
Hence, P(x)=xl and P(x)=0

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