Q A polynomial in x of degree greater than 3 leaves remainder 2, 6, 12 when divided - Maths - Complex Numbers and Quadratic Equations

Question

Q A polynomial in x of degree greater than 3 leaves remainder 2, 6,
12 when divided by x, (x— I) and
(x-2) respectively If R(x) be the remainder when polynomial is
divided by x(x - l) (x- 2), then

(A) R(x) is a quadratic polynomial (B) R (x) have real roots
(C) R (x) integer roots (D) R (10) = 132

Neha Sethi · Accepted Answer

Dear student
Given that:f0=2f1=6f2=12Let fx=pxxx-1x-2+ax2+bx+c  Now putting  f0=c⇒2=c    
(1)f1=a+b+c⇒6=a+b+c6=a+b+2    [using 1]a+b=4    a=4-b      
(2)f2=4a+2b+c12=4a+2b+c12=4(4-b)+2b+2    [using 1 and 2]12=16-4b+2b+26=2bb=3and from (2), we geta=4-3a=1  Hence , Rx=x2+3x+2 which is a quadratic polynomial
Regards