A factored polynomial reveals its roots, a key concept in understanding the behavior of these expressions.
Using the identity \( a^3 - b^3 = ( a -b ) ( a^2 + ab + b^2 ) \), which of the following must be a factor of
\[ x^3 + 8 ? \]
Using the identity \( a^2 - b^2 = (a-b) ( a+b) \), which of the following must be a factor of
\[ x^4 + 2500 ? \]
If \(a\) and \(b\) are positive numbers and \[x^4-38x^2y^2+y^4=(x^2+axy-y^2)(x^2-bxy-y^2),\] what is the value of \(a+b\)?
Suppose \(a\), \(b\), \(c\) and \(d\) are all non-positive integers such that the following is an algebraic identity in \(x\): \[(x-a)^2(x+a)^2=x^4+bx^3+cx^2+dx+256.\] What is the value of \(-a-b-c-d\)?