Question

let p(x) be a polynomial of degree more than 2 .when p(x) is divided by x-2, it leaves remainder 1 and when it is divided by x-3 , it leaves remainder 3. find remainder, when p(x) is divided by (x-2)(x-3).

Answer 1

remainder from (x-2) is 1
remainder from (x-3) is 3
now we have to find remainder after divided by (x-2)(x-3)
let quotient be Q, and remainder is 'R' of the format is ax+b
p(x)=Q*D+ax+b
p(1)=Q*0+2a+b
1=2a+b
p(2)=Q*0=3a+b
3=3a+b
on solving we get a=2, b=-3
p(x)=(x-2)(x-3)q(x)+(2x-3)
Hence remainder is 2x-3

Answer 2

remainder from (x-2) is 1

remainder from (x-3) is 3

now we have to find remainder after divided by (x-2)(x-3)

let quotient be Q, and remainder is 'R' of the format is ax+b

p(x)=Q*D+ax+b

p(1)=Q*0+2a+b

1=2a+b

p(2)=Q*0=3a+b

3=3a+b

on solving we get a=2, b=-3

p(x)=(x-2)(x-3)q(x)+(2x-3)

Hence remainder is 2x-3

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