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Advanced Factorization

Advanced factorization is a gateway to algebraic number theory, which mathematicians study in order to solve famous conjectures like Fermat's Last Theorem.

Factorization of Polynomials

The polynomial \[3x^5+13x^4+3x^3+3x^2+13x+3\] can be factorized as the product of two quadratics and one linear polynomial in \(x\), all with positive integer leading coefficients. What is the sum of all of the coefficients in the three factors?

Given that \(f(x) \) and \(g(x) \) are two non-constant polynomials with integer coefficients such that

\[ f(x) \times g(x) = x^6-6x^5+4x^4-12x^3-8,\]

evaluate \(\lvert f(2)+g(2) \rvert.\)

Details and assumptions

You may use the fact that \( f(2) \times g(2) = -168 \).

For what constant \(k\) can the polynomial \[(x+1)(x+5)(x+9)(x+13)+k\] be factorized into a perfect square of a quadratic in \(x?\)

If the polynomial \(x^4-9x^2+20\) is factorized as \((x+a)(x+b)(x+c)(x+d),\) where \(a, b, c\) and \(d\) are real numbers, what is the value of \(a^2+b^2+c^2+d^2?\)

Let \( f(x) = x^2 - 1 \). How many distinct real roots are there to \( f ( f( f(x))) = 0 \)?

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