Problema Solution
find the quadratic which has a remainder of -6 when divided by x-1, a remainder of -1 when divided by x-3 and no remainder when divided by x+1.
Answer provided by our tutors
let the quadratic we are looking for is ax^2 + bx + c, where a,b, c are constants that we need to find.
since
ax^2 + bx + c = (x-1)(ax + a + b) + a + b + c => a + b + c = -6
ax^2 + bx + c = (x-3)(ax + b + 3) + 3b + c + 9 => 3b + c + 9 = -1
ax^2 + bx + c = (x+1)(ax - a + b) + a - b + c => a - b + c = 0
by solving the system of equations
a + b + c = -6
3b + c + 9 = -1
a - b + c = 0
we find
a = -2
b = -3
c = -1
thus the quadratic equation is -2x^2 - 3x - 1 = 0.