Question

without actual division prove that

Answer 1

Let p(x) = 2x4 – 5x3 + 2x2 – x+ 2 firstly, factorise x2-3x+2.
Now, x2-3x+2 = x2-2x-x+2 [by splitting middle term]
= x(x-2)-1 (x-2)= (x-1)(x-2)
Hence, 0 of x2-3x+2 are land 2.
We have to prove that, 2x4 – 5x3 + 2x2 – x+ 2 is divisible by x2-3x+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 + 2 = 2x16-5x8+2x4+ 0 = 32 – 40 + 8 = 40 – 40 =0
Hence, p(x) is divisible by x2-3x+2.

Answer 2

x² - 3x + 2

= x² - 2x  - x + 2

= x(x-2) - 1(x-2)

= (x-1) (x-2)

x = 1 and 2

So let us check there are factors

2x⁴ - 5x³ - x + 2

if x =1

2 - 5 - 1 +2  = -2

No it not factor

If x =2

32 - 40 - 2 + 2 = -8

No it is not divisible x² - 3x + 2

I think your question is wrong. It can't prove it divisible by x² - 3x + 2