Find a quadratic polynomial which when divided by (x-1), (x-2), (x-3) leaves remainders 11, 22, 37 respectively.

Question

Find a quadratic polynomial which when divided by $(x-1)$, $(x-2)$, $(x-3)$ leaves remainders $11, 22, 37$ respectively.

Answer 1

Hint: Let $f(x)=ax^2+bx+c.$ Then by the given conditions, $$f(x)=(x-1)\phi_1(x)+11$$ $$f(x)=(x-2)\phi_2(x)+22$$ $$f(x)=(x-3)\phi_3(x)+37,$$ for some polynomials $\phi_1(x),\phi_2(x),\phi_3(x)$ of degree $1$.

Answer 2

Starting point: write down what it means for a quadratic polynomial $P$ to leave remainder $11$ when divided by $x-1$...