A polynomial p(x) when divided by (x-1), leaves the remainder 2, the same polynomial when divided by x-3, leaves the remainder 1. If same polynomial is divided by (x-1)(x-3) what will be remainder?
1Designate the unknown polynomial by p(x) and let q(x) designate the quotient and r(x) = ax+ b the remainder resulting from division of p (x) by (x-1)(x-3)
then
p(x) = (x-1)(x-3)q(x) + ax+ b ...........(1)
by conditions of problem
p(x) = (x-1)q1(x) + 2 whence p (1) = 2
p(x) = (x-3)q2(x) + 1 whence p (3) = 1
if we substitute x = 1 and x = 3 successively in 1 we obtain
2 = p(1) = a+b
1 = p(3) = 3a+b
from which we obtain
a = - 12
b = 52
therefore remainder is -12x + 52
36@ninepointcircle ->
Since, the degree of the polynomial is not given so how can you assume the
remainder to be ax + b?........
1@rahulmishra
we know that the degree of r(x) i.e the remainder is always less than the divisor
since (x-1)(x-3) has degree 2 ...r(x) can have a degree 1 or be a constant and therefore we chose ax+b which is a linear polynomial.
1@rahulmishra
http://www.targetiit.com/iit-jee-forum/posts/gmo-problems-18102.html
in this post u said about a book on Olympiads sent to u by nishant sir ..can u pls tell me the title and publisher of the book
36@ninepointcircle -> post ur id in my chatbox and i will send it to you....!
I am in xth