<s>[INST] Dividing polynomial by binomial using remainder theorem An A level text book claims that one can find the quotient by first:
1.) Setting up an identity, $f(x)≡ Q(x)(divisor) + remainder$
2.) Finding the coefficients
However, another A level text book says, "Note. This theorem gives a (simple) method for evaluating the remainder only. If the quotient is required, long division must be used."
The question is: Divide $x^3 + x^2 - 7$ by $x-3$ using the remainder theorem. 
In this example,
1.) They set up the identity: $x^3 + x^2 - 7 ≡ (Ax^2 + Bx + C)(x-3) + D$
2.) They let $x=3$ to find coefficient $D$ 
3.) They let $x=0$ to find coefficient $C$
4.) To find coefficients $A$ and $B$, the text book then goes on to "comparing the coefficients". No more detail is given as to how "comparing the coefficients" to find $A$ and $B$ is achieved. 
Can you find coefficients $A$ and $B$ using this method ONLY? If so, how?
 [/INST] Assuming the equation was meant to be:
$$x^3+x^2-7 = (Ax^2+Bx+C)(x-3)+D$$
Setting $x=3$, we sse that $3^3+3^2-7 = 29 = D$.
Setting $x=0$, we see that $-7 = D-3C=29-3C$, so $C=12$.
Setting $x=1$, we see that $-5=(A+B+12)(-2)+29$, or $A+B+12 = 17$.
Setting $x=2$, we see that $5 = (4A+2B+12)(-1)+29 = 4A+2B+12 = 24$.
So now you have two linear equations for $A,B$, and you can solve those.
(Definitely redo my arithemetic, it could be in error.)
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