Number Theory (Polynomials) Find The Remainder? A polynomial $f(x) = x^{50} + 3x^{49} + 3x + 12$ when divided by $x - a$,  it leaves remainder $3$ & when its quotient is further divided by $x - b$ it leaves remainder $5$, also when $f(x)$ is divided by $x^2 - ( b + a)x + ab$ the remainder is $x + 6$. Find $a$?

$P=x^{50}+3x^{49}+3x+12=x^{49}(x+3)+3(x+3)+3=k(x-a)+3$
So the quotient k is:
$k=\frac{(x^{49}+3)(x+3)}{x-a}$
⇒$\frac{(x^{49}+3)(x+3)}{x-a}=k_1(x-b)+5$
⇒$(x^{49}+3)(x+3)=k_1(x-a)(x-b)+5(x-a)$
⇒$p=(x^{49}+3)(x+3)+3=k_1(x-a)(x-b)+5(x-a)+3$
Therefore:
$5(x-a)+3=x+6$ ⇒ $a=\frac{4x-3}{5}$
