Suppose the polynomial $P(x)$ with integer coefficients satisfies the following conditions:
(A) If $P(x)$ is divided by $x^2 − 4x + 3$, the remainder is $65x − 68$.
(B) If $P(x)$ is divided by $x^2 + 6x − 7$, the remainder is $−5x + a$.
Then we know that $a =$?
I am struggling with this first question from the 1990 Japanese University Entrance Examination. Comments from the linked paper mention that this is a basic application of the "remainder theorem". I'm only familiar with the polynomial remainder theorem but I don't think that applies here since the remainders are polynomials. Do they mean the Chinese remainder theorem, applied to polynomials?
So for some $g(x)$ and $h(x)$ we have:
$$P(x) = g(x)(x^2-4x+3) + (65x-68),\\
P(x) = h(x)(x^2+6x-7) + (-5x+a),$$
which looks to have more unknowns than equations. How should I proceed from here?
Best Answer
You need only $P(1)$, since
Hence,