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Advanced Factorization

Factorization of Cubics


If aa, bb and cc are real numbers such that a+b+c=0,abc=14,a+b+c=0, abc=-14, what is the value of a2(b+c)+b2(c+a)+c2(a+b)?a^2(b+c)+b^2(c+a)+c^2(a+b)?

For a linear expression f(x)=ax+bf(x)=ax+b, the polynomial g(x)=x3x2+2f(x)g(x) = x^3-x^2+2f(x) can be factorized as (x1)(x+α)(x+β)(x-1)(x+\alpha)(x+\beta). If αβ=22\alpha\beta=22, what is the value of f(12)f(12)?

xx is a number such that x2+5x+25=0x^2+5x+25=0. What is the value of x3x^3?

What is the largest positive integer aa such that the cubic polynomial x321x2+(20+a)xax^3-21x^2+(20+a)x-a can be factorized as the product of 33 distinct linear polynomials?

If aa is a real number and a cubic equation f(x)=x318x2+(a+32)x2af(x) = x^3-18x^2+(a+32)x-2a has a repeated root, what is the sum of all the possible values of aa?


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