Algebra

Polynomial Factoring

Factoring Polynomials of Higher Degree

         

The polynomial x47x376x27x+1x^4-7x^3-76x^2-7x+1 is factorized as (x2+ax+b)(x2+cx+d),(x^2+ax+b)(x^2+cx+d), where aa, bb, cc and dd are real numbers and a>0a>0. What is the value of adbcad-bc?

The polynomial x55x431x331x25x+1x^5-5x^4-31x^3-31x^2-5x+1 is factorized as (x+k)(x2+ax+b)(x2+cx+d),(x+k)(x^2+ax+b)(x^2+cx+d), where aa, bb, cc, dd and kk are constants and a>0a>0. What is the value of (a+k)dbc(a+k)d-bc?

Suppose f(x,y)f(x,y) and g(x,y)g(x,y) are non-constant polynomials with integer coefficients in two variables, xx and yy, such that f(x,y)g(x,y)=x33y3+3x2xy3f(x,y)\cdot g(x,y)=x^3-3y^3+3x^2-xy^3 Find f(3,3)+g(3,3).|f(3,3)|+|g(3,3)|.

If aa, bb, cc, dd and ee are non-negative integers such that

x6289x4x2+289=(xa)(x+b)(xc)(x+d)(x2+ex+1),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+ea+b+c+d+e?

If aa and bb are positive numbers satisfying x45x3+7x243x+40=(xa)(xb)(x2+x+c),\begin{aligned} & x^4-5x^3+7x^2-43x+40 \\ &= (x-a)(x-b)(x^2+x+c), \end{aligned} what is the value of a+b+ca+b+c?

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