Find a polynomial of degree 3 such that when divided by x^2-3x has a remainder of 6x-5 and when divided by x^2-5x+8x has a remainder of 2x-7

Question

Find 3rd degree polynomial given divisor and remainder (function in x)

Answer 1

Is this problem stated correctly? The remainder theorem with the divisor and remainder as stated is:

p(x) = (x²-3x)q₁(x) + (6x-5)

Since the zeroes of (x²-3x) are x=0 and x=3,

p(0) = -5, and p(3) = 13,

likewise for the other divisor,

p(x) = (x²-5x+8x)q₂(x) + (2x-7)

and since (x²-5x+8x) = x²+3x, the zeroes are x=0 and x=-3

p(0) = -7 and p(-3) = -13

According to the problem as stated, p(x) has two different values (-5 and -7) at x=0 which is not possible.