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Newton's theorem states that if a polynomial f(x) of degree n has n distinct roots, then:

1. The sum of the roots is equal to the coefficient of x^(n-1) divided by the coefficient of x^n.

2. The sum of the products of every possible pair of roots is equal to the coefficient of x^(n-2) divided by the coefficient of x^n.

3. The sum of the products of every possible triplet of roots is equal to the coefficient of x^(n-3) divided by the coefficient of x^n.

And so on, until the last statement:

n. The product of all the roots is equal to the constant term divided by the coefficient of x^n.

This theorem can be proved using Vieta's formulas, which state that the sum of the roots of a polynomial is equal to the negative of the coefficient of the second-to-last term divided by the coefficient of the last term, and the product of the roots is equal to the constant term divided by the coefficient of the last term.

To prove the first statement of Newton's theorem, we can use the fact that the sum of the roots is equal to the negative of the coefficient of the second-to-last term divided by the coefficient of the last term. Let the roots be r1, r2, ..., rn. Then we have:

r1 + r2 + ... + rn = -a_(n-1)/a_n

Multiplying both sides by -a_n, we get:

-a_n(r1 + r2 + ... + rn) = a_(n-1)

Dividing both sides by -a_n, we get:

r1 + r2 + ... + rn = a_(n-1)/a_n

This proves the first statement of Newton's theorem. The other statements can be proved in a similar way, using Vieta's formulas and the fact that the sum of the products of every possible pair of roots is equal to the negative of the coefficient of the third-to-last term divided by the coefficient of the last term, and so on.

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