Prove an nth-degree polynomial has exactly n roots

• I
The attachment below proves that an nth-degree polynomial has exactly ##n## roots.

The outline of the proof is as follows:
Suppose (1.1) has ##r## roots. Then it can be written in the form of (1.8) by factor theorem.
Next use the second fundamental result in algebra (SFRA): if ##f(x)=F(x)## for all values of ##x##, then their coefficients ##a_i##'s in (1.1) are the same. So we have [after expanding (1.8)] ##Ax^r=a_nx^n##. Thus ##r=n##.

The last sentence suggests that if ##f(x)## can be written in the form (1.8) then the proof is complete. And it can be by factor theorem.

My issue with the proof is that the condition for SFRA is not satisfied and hence we cannot compare the coefficients ##a_i##'s. We do not know that ##f(x)=F(x)## for all values of ##x##; we only know ##f(x)=F(x)## for ##r## values of ##x##, for those ##r## roots ##\alpha_i##'s.

So how could we justify the use of SFRA?

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jbunniii
Homework Helper
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The last sentence suggests that if ##f(x)## can be written in the form (1.8) then the proof is complete. And it can be by factor theorem.
If you have established that ##f(x)## can be written in the form (1.8), then doesn't that mean precisely that ##f(x) = F(x)## for all ##x##? If not, then what does it mean?

If you have established that ##f(x)## can be written in the form (1.8), then doesn't that mean precisely that ##f(x) = F(x)## for all ##x##? If not, then what does it mean?

Oh, this is a good point!

Then I've not established that ##f(x)## can be written in the form (1.8) with ##A## being a constant, but only that ##f(x)=g(x)(x-\alpha_1)(x-\alpha_2)...(x-\alpha_r)## where ##g(x)## is a polynomial.

Wikipedia: "In spite of its name, there is no purely algebraic proof of the theorem, since any proof must use the completeness of the reals (or some other equivalent formulation of completeness), which is not an algebraic concept." https://en.wikipedia.org/wiki/Fundamental_theorem_of_algebra

Does it mean there is no elementary proof for this theorem?

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jbunniii