Find a polynomial of degree 3 with rational coefficients, which divided by $X^2 - 5x + 6$ has the remainder $2x - 1$, and at the division with $X^2 + 1$ has the remainder $x - 2$.
closed as off-topic by Jack D'Aurizio, hardmath, user63181, Ahaan S. Rungta, Grigory M Jan 12 '15 at 17:37
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Let the cubic polynomial be
$$(ax+b)(x^2-5x+6)+2x-1=(cx+d)(x^2+1)+x-2$$ where $a,b,c,d$ are arbitrary constants
Compare the constants & the coefficients of $x,x^2,x^3$ to find $a,b,c,d$
For $x=2$ result $3=5(2m+n)$
For $x=3$ result $5=10(3m+n)+1$...