Question

Without actual division, prove that 2x ​4 -5x 3 +2x 2 -x+2 is divisible by x 2 -3x+2.

Answer 1

x2-3x+2 = (x-2)(x-1)
P(x)=2x ​4 -5x 3 +2x 2 -x+2 is divisible by x 2 -3x+2  is equivalent to 
1 and 2 are zeros for 2x ​4 -5x 3 +2x 2 -x+2 

we can verify thatP(1) = 0
and P(2)=0

then
2x ​4 -5x 3 +2x 2 -x+2 is divisible by x 2 -3x+2