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Polynomial Factoring

A factored polynomial reveals its roots, a key concept in understanding the behavior of these expressions.

Compound Quadratics

         

If \(a\), \(b\), \(c\) and \(d\) are positive numbers that satisfy \[x^4-40x^2+4 = (x^2+ax-b)(x^2-cx-d),\] what is the value of \(a+b+c+d\)?

If \(a\) and \(b\) are positive numbers such that \[x^3+2x^2y-9x-18y=(x+a)(x-b)(x+cy),\] what is the value of \(a+b+c\)?

If \(b\) and \(c\) are positive numbers such that \[(x^2-1)(x^2+8x+7)=(x+a)^2(x+b)(x-c),\] what is the value of \(a+b+c\)?

If \(a\) and \(b\) are positive integers such that \[\begin{align} & x^2 y^2-9x^2z^2-10xy^2+90xz^2+25y^2-225z^2\\ & =(x-a)^2(y-bz)(y+bz), \end{align}\] what is the value of \(a+b\)?

A polynomial \(x^4-226x^2+225\) can be factorized as \[(x-a)(x-b)(x-c)(x-d).\] If the four constants satisfy \(a < b < c < d\), what is the value of \(bc-ad\)?

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