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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 Cubics


If \(a\), \(b\) and \(c\) are real numbers such that \[a+b+c=0, abc=-14,\] what is the value of \[a^2(b+c)+b^2(c+a)+c^2(a+b)?\]

For a linear expression \(f(x)=ax+b\), the polynomial \[g(x) = x^3-x^2+2f(x)\] can be factorized as \((x-1)(x+\alpha)(x+\beta)\). If \(\alpha\beta=22\), what is the value of \(f(12)\)?

\(x\) is a number such that \(x^2+5x+25=0\). What is the value of \(x^3\)?

What is the largest positive integer \(a\) such that the cubic polynomial \[x^3-21x^2+(20+a)x-a\] can be factorized as the product of \(3\) distinct linear polynomials?

If \(a\) is a real number and a cubic equation \[f(x) = x^3-18x^2+(a+32)x-2a\] has a repeated root, what is the sum of all the possible values of \(a\)?


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