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Without actual division prove that 2x^4-...

Without actual division prove that `2x^4-6x^3+3x^2+3x-2` is exactly division by `x^2-3x+2.`

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`P(x)=2x^4−5x^3+2x^2−x+2` and
`x^2−3x+2=(x−1)(x−2)`
Let `x^2−3x+2` is the factor of `P(x)`
Then, `x^2−3x+2=0`
`⇒(x−1)(x−2)=0`
`⇒x=1` and `2`
For `x=1`
`P(1)=0`
For `x=2`
`P(2)=0`
`∵P(1)=P(2)=0`
`∴P(x)` is divisible by `x^2−3x+2`
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