Advanced Factorization

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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