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## Homework Statement

Prove that if a Polynomial P(x) of degree n, shares n points with x

^{n}, then P(x)=x

^{n}.

(a more general proof would be almost the same but for two polynomials, but I think I've proved this if this is proved)

## Homework Equations

(FTA) Fundamental Theorem of Algebra - Every P(x) of degree n has at most n roots.

## The Attempt at a Solution

An idea was to make an analogy between the proof of FTA and this probem. To use for example the factorization: P(x)= ∏(x-r

_{i}) Which when proved valid, proves the FTA too. But I don't understand the proofs given online too well.

But also if we accept FTA, we could also possibly analyze P(x)-x

^{n}which shall be called D(x).

D(x) has degree (n-1) since x

^{n}-x

^{n}=0.

Also since P(x) and x

^{n}have n equal points, then D(x) has n zeroes.

But FTA says that the D(x) of degree (n-1) has at most n-1 zeroes. But it has n zeroes. Therefore it must equal 0 itself. P(x)-x

^{n}=0 ⇔ P(x)=x

^{n}

Is this correct? Is there any simpler way to prove or address this or the more general problem?

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