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Polynomial Factoring

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

Higher Degree Polynomials

         

The polynomial \[x^4-7x^3-76x^2-7x+1\] is factorized as \[(x^2+ax+b)(x^2+cx+d),\] where \(a\), \(b\), \(c\) and \(d\) are real numbers and \(a>0\). What is the value of \(ad-bc\)?

The polynomial \[x^5-5x^4-31x^3-31x^2-5x+1\] is factorized as \[(x+k)(x^2+ax+b)(x^2+cx+d),\] where \(a\), \(b\), \(c\), \(d\) and \(k\) are constants and \(a>0\). What is the value of \((a+k)d-bc\)?

Suppose \(f(x,y)\) and \(g(x,y)\) are non-constant polynomials with integer coefficients in two variables, \(x\) and \(y\), such that \[f(x,y)\cdot g(x,y)=x^3-3y^3+3x^2-xy^3\] Find \(|f(3,3)|+|g(3,3)|.\)

If \(a\), \(b\), \(c\), \(d\) and \(e\) are non-negative integers such that

\[x^6-289x^4-x^2+289=(x-a)(x+b)(x-c)(x+d)(x^2+ex+1),\]

what is the value of \(a+b+c+d+e\)?

If \(a\) and \(b\) are positive numbers satisfying \[\begin{align} & x^4-5x^3+7x^2-43x+40 \\ &= (x-a)(x-b)(x^2+x+c), \end{align}\] what is the value of \(a+b+c\)?

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