You did well in identifying the possible rational roots. Like you, I checked out x=3 as a possible root. The 3 coefficients remaining after that are 1, -4 and 2.
They correspond to the quadratic expression 1x^2 -4x + 2.
We can apply the quadratic formula here. a=1, b=-4 and c=2.
-(-4) plus or minus sqrt( [-4]^2 - 4(1)(2) ) Then x = ------------------------------------------------------------ 2(1) 4 plus or minus sqrt(8) 4 plus or minus 2sqrt(2) or x = ----------------------------------- = ----------------------------------- 2 2
Then x = 2 plus or minus sqrt(2).
If you wanted, you could check your work here by using synthetic division. Let the divisor be 2+sqrt(2) and the 3 coefficients be 1 -4 2.
