# Find a polynomial of degree 3 with rational coefficients, given remainders mod two quadratics [closed]

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 MJan 12 '15 at 17:37

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• You should indicate what motivates this Question, or what you've tried to solve it, to help Readers understand your level of understanding and respond cogently. – hardmath Jan 12 '15 at 16:45
• @hardmath you are right, but I was lucky enough and the second answer is exactly what I needed. About my level of understanding, we just learned about polynomials 4 days ago in highschool. – cristid9 Jan 12 '15 at 16:57

HINT

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

$$\iff (a)x^3+x^2(b-5a)+x(2+6a-5b)+6b-1=(c)x^3+x^2(d)+x(c+1)+d-2$$

Compare the constants & the coefficients of $x,x^2,x^3$ to find $a,b,c,d$

$$P(X)=(aX+b)(X^2-5X+6)+2X-1=(mX+n)(X^2+1)+X-2$$

For $x=2$ result $3=5(2m+n)$

For $x=3$ result $5=10(3m+n)+1$...