Grade 12th passAlgebra

If a polynomial P(x) is divided by x-a, x-b, x-c, we get a,b,c as remainders respectively (x€R). Then what will be the remainder when P(x) is divided by (x-a)(x-b)(x-c)?

Akash Sen

9 Years agoGrade 12th pass

2 Answers

Harsh Patodia

9 Years ago

Using Remiander theorem
f(a) is the remainder when f(x) is divided by x-a
when f(x) is divided by q(x) the remainder will be having degree less than q(x) so when P(x) is divided by (x-a),(x-b),(x-c) the remainders are P(a),P(b) and P(c) respectively
therefore P(x)=(x-a)(x-b)(x-c)g(x) + (px^2+qx+r) where (px^2+qx+r)is remainder g(x) is the quotient
P(a)=pa^2+qa+r=a --- (I)
P(b)=pb^2+qb+r=b----(II)
P(c)=pc^2+qc+r=c-----(III)
Closely observing u can q=1 p and r =0
So remiander will be x
and Secondly you could have also solved equations by using II-I and III-I.

mycroft holmes

9 Years ago

Note that P(a) =a, P(b) = b, and P(c) = c.

Hence if we consider the polynomial Q(x) = P(x) – x, we see that Q(a) = Q(b) = Q(c) = 0. By Factor theorem we conclude (x-a)(x-b)(x-c) is a factor of Q(x).

i.e. Q(x) = (x-a)(x-b)(x-c) R(x) for some polynomial R(x).

so that P(x) – x = (x-a)(x-b)(x-c) R(x) or

P(x) = (x-a)(x-b)(x-c) R(x)+x

From the above eqn is evident that when P(x) is divided by (x-a)(x-b)(x-c), the remainder is x