Find a monic quadratic polynomial, f(x), which divides bothg(x)=12x3−30x2+18x−12andh(x)=6x4+3x3+6x2+3Problem statement:
Find a monic quadratic polynomial, f(x), which divides both
g(x)=12x3−30x2+18x−12
and
h(x)=6x4+3x3+6x2+3
My take on it:
I divided h(x) by g(x) to get the quotient and remainder such that
6x4+3x3+6x2+3=(12x3−30x2+18x−12)(12x+32)+3(14x2−11x+7)
It is also the case that any polynomial divisor of both g(x) and h(x) must also divide the remainder polynomial when h(x) is divided by g(x).
So following on from that, our common factor for g(x) and h(x) that we're trying to find, would also have to be a factor of our remainder, 3(14x2−11x+7). Yet the remainder cannot be factorised any further to turn it into a monic quadratic polynomial.