Question

Xác định a, b, c để:

x^3 - ax^2 + bx - c = (x - a) (x - b) (x - c) với mọi x.

Answer 1

Ta có:

\(\left(x-a\right).\left(x-b\right).\left(x-c\right)\)

\(=x^3-\left(a+b+c\right).x^2+\left(ab+bc+ca\right).x-abc\)

\(\Leftrightarrow\left[{}\begin{matrix}a+b+c=a\\ab+bc+ca=b\\a.b.c=c\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}b+c=0\left(1\right)\\ab+bc+ca-b=0\left(2\right)\\c.\left(ba-1\right)=0\left(3\right)\end{matrix}\right.\)

Xét \((3)\) ta có :

\(\Leftrightarrow\left[{}\begin{matrix}c=0\\a.b=1\end{matrix}\right.\)

Với \(c=0\) thì \(b=0\) ; \(a\) tùy ý

Với \(a.b=1\) thì:

\(\Leftrightarrow\left[{}\begin{matrix}a=-1\\b=-1\end{matrix}\right.\)