Question

Biết đa thức f(x) chia cho x-2 dư 11, chia cho x-3 dư 23. Hỏi f(x) chia cho x^2 - 5x + 6 dư bao nhiêu?

Answer 1

\(f\left(x\right)\) chia \(x-2\) dư \(11\Leftrightarrow f\left(x\right)=\left(x-2\right)H\left(x\right)+11\Leftrightarrow f\left(x\right)-11=\left(x-2\right)H\left(x\right)\)

\(f\left(x\right)\) chia \(x-3\) dư \(23\Leftrightarrow f\left(x\right)=\left(x-3\right)G\left(x\right)+23\Leftrightarrow f\left(x\right)-23=\left(x-3\right)G\left(x\right)\)

Do vậy \(\left(f\left(x\right)-11\right)\left(f\left(x\right)-23\right)=\left(x-2\right)\left(x-3\right)H\left(x\right)G\left(x\right)\)

\(\Rightarrow f\left(x\right)^2-34f\left(x\right)+253⋮\left(x-2\right)\left(x-3\right)\)

Do vậy \(f\left(x\right)\) chia \(\left(x-2\right)\left(x-3\right)\) dư \(-253\)