Given \(P\) is a polynomial that is not constant, satisfy \( P\left( P(x) \right) = (x^2 + x +1) \times P(x) \) for every real \(x\). Find \( P(10) \).

Hint: Let \(d\) denote the degree of polynomial \(P(x) \), then the degree of LHS is \(2d\), and the degree of RHS is \(d+2\). So can you solve for \(d\)? And can you determine the function \(P(x) \)?

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## Comments

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TopNewestHint:Let \(d\) denote the degree of polynomial \(P(x) \), then the degree of LHS is \(2d\), and the degree of RHS is \(d+2\). So can you solve for \(d\)? And can you determine the function \(P(x) \)?Log in to reply

I got \( P(x) = x^2 + x\) but I dont know, maybe there's another possible formula for \( P(x) \)

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No, there isn't. If you try for (constant of quadratic polynomial not equal 0), then you get a system of equations that has no solution.

I know that this isn't the best/fastest way to get the answer, but it works!

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