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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 Quadratics

         

If \(x-y-5=0\), then the polynomial \[3x^2+xy-y^2-25x-y+50\] can be simplified as \(ay(y+b)\). What is \(a+b\)?

Given that \[\begin{align} & x = -a+b+c, \\ & y = a-b+c, \\ & z = a+b-c ,\\ & a^2+b^2+c^2 =20, \end{align}\] find the value of \(x^2+y^2+z^2+xy+yz+zx\).

Suppose that \(a\) is an integer such that \[x^2+2xy-80y^2+ax+18y+81\] can be factorized as the product of linear expressions in \(x\) and \(y\). What is the value of \(a\)?

\(a, b\) and \(c \) are integers such that \[ (x-a)(x-20) + 1 = (x+b)(x+c) .\] What is the sum of all possible values of \( a \)?

Suppose \(a\), \(b\) and \(c\) are real numbers such that \[a+b+c=5.\] If \[x=a-2b+3c, y=b-2c+3a, z=c-2a+3b,\] what is the value of \[(x^2+2xy+4)+(y^2+2yz+4)+(z^2+2zx+4)?\]

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