Algebra
Polynomial Factoring
Find the sum of all real solutions of the equation \[x^3+x^2 + x=8x^2+8 x + 8.\]
The polynomial in two variables
\[ f(x,y) = xy + 3y - 4x -12 \]
can be expressed as the product of two linear factors with integer coefficients, namely
\[ f(x,y) = (ax + by + c) ( dx + ey + f). \]
Evaluate \( |a+b+c| + |d+e+f| \).
If \(a, b, c\) and \(d\) are positive numbers and \[5x^3-45xy^2+4x^2-36y^2=(ax+b)(x+cy)(x-dy),\] what is the value of \(a+b+c+d\)?
If \(a, b, c,\) and \(d\) are positive numbers such that \[\begin{align} & (x^2-5x)^2-18x^2+90x-144 \\ &= (x-a)(x-b)(x-c)(x+d), \end{align} \] what is the value of \(a+b+c+d\)?
If \(a\) and \(b\) are positive numbers such that \[x^3+5x^2y-9x-45y=(x+a)(x-b)(x+cy),\] what is the value of \(a+b+c\)?