SOLUTION: If a quadratic polynomial is divided by (x-2) and (x+1) leave the remainder 11 and 1 respectively, find the polynomial.

    If a quadratic polynomial with the leading coefficient of 1, when divided by (x-2) and (x+1) 
    leaves the remainder 11 and 1 respectively, find the polynomial.

Let  P(x) =  be this quadratic polynomial with unknown coefficients "b" an "c".


The fact that  P(x) leaves the remainder 11 when divided by (x-2) is equivalent (according to the "Remainder Theorem") that

P(2) = 11,   or    = 11.        (1)


The fact that  P(x) leaves the remainder 1 when divided by (x+1) is equivalent (according to the same theorem) that

P(-1) = 1,   or    = 1.     (2)


Thus we have this system of two equations to find the coefficients "b" and "c":

2b + c = ,    or   2b + c = 7,     (1')

-b + c = ,   or   -b + c = 0     (2')



To solve the system, subtract eq(2') from eq(1')  (both sides). You will get

2b - (-b) = 7 - 0,   or  3b = 7  ====>  b = .


Then from eq(2')  c = b = .


Thus the quadratic polynomial is  P(x) = .


Check.  P(2) =  =  =  = 4 + 7 = 11,   and

        P(-1) =  =  = 1.   ! Correct !


Answer.  The polynomial is  P(x) = .