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Verify division algorithm for the polyno...

Verify division algorithm for the polynomials
`p(x)=3x^(4)-4x^(3)-3x-1andg(x)=x-2`.
Find `p(2)`. What do you observe ?

Text Solution

Verified by Experts

By long division , we have

`therefore` quotient , `q(x)=3x^(3)+2x^(2)+4x+5` and remainder , `r(x)=9`.
Now , `g(x)xxq(x)+r(x)=(x-2)(3x^(3)+2x^(2)+4x+5)+9`
`=3x^(4)-4x^(3)-3x-10+9`
`=3x^(4)-4x^(3)-3x-1=p(x)`.
`thereforep(x)=g(x)xxq(x)+r(x)`, where degree `r(x)=0lt1=` degree g(x).
Thus , division algorithem is verified.
Now `p(2)=3xx2^(4)-4xx2^(3)-3xx2-1=48-32-6-1=9`.
Observation When p (x) is divided by (x-2) , then the remainder is p(2).
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