Algebra

# 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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