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A certain polynomial P(x),x E R WHEN DIVIDED BY (x-a),(x-b) and (x-c) leaves remainder a,b,c.then find the remainder when P(x) is divided by (x-a)(x-b)(x-c) where a,b,c are distinct

A certain polynomial P(x),x E R WHEN DIVIDED BY (x-a),(x-b) and (x-c) leaves remainder a,b,c.then find the remainder when P(x) is divided by (x-a)(x-b)(x-c) where a,b,c are distinct

Grade:12

2 Answers

Sunil Raikwar
askIITians Faculty 45 Points
7 years ago
since divisor is cubic therefore degree of remaider 2 or less then 2. let remainder be Ax2+Bx+C
p(x)= (x-a)(x-b)(x-c).q(x)+Ax2+Bx+c [dividend= divisor x quotient + remainder]............1
put x=a,b,c in 1
p(a)= Aa2+Ba+C or a=Aa2+Ba+C.........2
p(b)= Ab2+Bb+C or b= Ab2+Bb+C..............3
p(c)=Ac2+Bc+c or c=Ac2+Bc+C.............4
subtract equation 2 from 3 & 3 from 4, find the value of A,B and C and get the required answer.


Thanks & Regards
Sunil Raikwar
askIITiann's faculty

mycroft holmes
272 Points
7 years ago
Consider the cubic polynomial Q(x) = P(x) - x. This polynomial has its roots as a,b,c. Hence P(x) - x = (x-a)(x-b)(x-c) R(x) for some polynomial R(x). Hence P(x) = (x-a)(x-b)(x-c) R(x)+x We can immediately see that the remainder when P(x) is divided by (x-a)(x-b)(x-c) is x

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