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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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