what is the factor of x^6+4xy-x+3xy^2+y-2?

Question

factoring by regrouping

Answer 1

x(x⁵-1+y(4+3y))+y-2 is one way to regroup, but it doesn't help to factorise.

However:

When y=0, x⁶-x-2=(x+1)(x⁵-x⁴+x³-x²+x-2);

when y=-1, x⁶-2x-3=(x+1)(x⁵-x⁴+x³-x²+x-3);

when y=-2, x⁶+3x-4=(x-1)(x⁵+x⁴+x³+x²+x+4).

Other values of y do not produce a polynomial with rational factors. But:

When y=x, x⁶+3x³+4x²-2=(x+1)(x⁵-x⁴+x³+2x²+2x-2).