how to find the remainder when a polynomial $p(x)$ is divided my another polynomial $q(x)$

Question

i was solving the question from the book IIT FOUNDATION AND OLYMPIAD - X and i was solving the problems of polynomials-III. so on the page number 136, there is a question (question 17) given below:

The remainder when $x$^100 is divided by $x^2-3x+2$ is:

a) $(2$^100$-1)x + (-2$^100$ +2) $

b) $(2$^100$+1)x + (-2$^100$ -2) $

c) $(2$^100$-1)x + (-2$^100$ -2) $

d) none

as far as i tried to find the remainder, i tried long division method but it was getting more and more complicated, then i used systematic method of division but i can't get the corret option what is the correct option. please explain me how did you find the remainder. thanks

and yes its answer is option (a)

score 4 · Accepted Answer · answered Jun 21 '14 at 15:27

4

Hint Write $x^{100}= (x^2-3x+2)q(x) + ax+b$. Now plug $x=1$ and $x=2$ to find $a$ and $b$.

answered Jun 21 '14 at 15:27

lhf

216,483

what will i get after finding the value of a and b ? – anni Jun 21 '14 at 15:31
1

@anni, the remainder is $ax+b$. – lhf Jun 21 '14 at 15:31
okey i got it... thanks – anni Jun 21 '14 at 15:32

Bill Dubuque · Answer 2 · 2014-06-21T15:32:26.740

3

Hint $\,\ f\equiv g\ \ ({\rm mod} (x\!-\!1)(x\!-\!2))\ \iff x\!-\!1,x\!-\!2\mid f\!-\!g\ \iff f(1) = g(1),\ f(2) = g(2)$

edited Jun 21 '14 at 15:32

answered Jun 21 '14 at 15:23

Bill Dubuque

272,048

should it go for none option ? – anni Jun 21 '14 at 15:28
2

@anni By above, with $,f(x) = x^{100},,$ an answer $,g(x),$ is correct $\iff g(1) = 1,\ g(2) = 2^{100}.\ \ $ – Bill Dubuque Jun 21 '14 at 15:34

how to find the remainder when a polynomial $p(x)$ is divided my another polynomial $q(x)$

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