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

Explanation

Let `p(x) =2x^(4) -5x^(3)+2x^(2)-x+2`firstly,factorise `x^(2)-3x+2.`
now `x^(2)-3x+2=x^(2)-2x-x+2` [by splitting middle term]
`=x(x-2)-1(x-2)=(x-1)(x-2)`
hence `o. "of " x^(2)-3x+2 are 1 and 2.`
we have to prove that ` 2x^(4)-5x^(3)+2x^(2) -x+2` is divisible by`x^(2)-3x+2`
i.e ., to prove that p(1) =0 and p(2) -0
Now , `p(1) =2(1)^(4)-5(1)^(3)+2(1)^(2)-1+2`
` =2-5+2-1+2=6-6=0`
and `p(2) =2(2)^(4)-5(2)^(3)+2(2)-2+2`
`=2xx16-5xx8+2xx4+0`
`=32-40+8=40-40=0`
hence p(x) is divisible by` x^(2)-3x+2.`