(Q12)On dividing x3-3x2+x+2 by a polynomial g(x), the quotient and remainder were x-2 and -2x+4, respectively.Find g(x)

Answer

Giuven p(x)=x3-3x2+x+2

q(x)=x-2 and

r(x)= -2x+4

By divison algorithm we know that

Dividend = Divisor×Quotient+Remainder

p(x)=q(x)×g(x)+r(x)

Therefore x3-3x2+x+2

=(x-2)×g(x)+(-2x+4)

x3-3x2+x+2+2x-4=(x-2)×g(x)

g(x)=
x3-3x2+3x-2
x-2

on dividing x3-3x2+3x-2 by x-2 we get

2-+
x-2) 3 -x2 + x -
3 - x2
-   +
-2 + x -
-2 + x
+   -
-
-
-  +
0

Hence g(x) = 2 - +