Question

What must be added to p(x) = 4x4+2x3-2x2+x-1 so that the resulting polynomial is divisible by f(x) x2+2x-3

Answer 1

since f(x)-r(x)=g(x)*q(x)

therfore  we will divide 4x4+2x3-2x2+x-1 by x2+2x-3 with the help of division algorithm method

on dividing this we will get quotient as = 4x2-6x+22

n we will get remainder as = -61x+65

hence we should add r(x) to f(x) so that resulting polynomial is divisible by g(x)

Answer 2

By division algorithm, we have  

f(x)=g(x)×q(x)+r(x)

⇒f(x)−r(x)=g(x)×q(x)                      ⇒f(x)+−r(x)=g(x)×q(x)

Clearly, RHS is divisible g(x). Therefore LHS is also divisible by g(x). Thus, if we add −r(x) to f(x), then the resulting polynomial is divisible by g(x) Let us now find the remainder when f(x) is divided by g(x)

∴r(x)=−61x+65                                    −r(x)=61x−65

Hence we should add −r(x)=61x−65 to f(x) so that the resulting polynomial is divisible by g(x).