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 ∀k≥0
So far: If α is a root of P(x) then so is (α+1)2k-1 for all k≥1
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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