Grade 12Algebra

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

chaman

12 Years agoGrade 12

2 Answers

Sunil Raikwar

12 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

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mycroft holmes

12 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