Question

: Xác định hệ số a, b, c biết:

a) ( x² + cx + 2)( ax + b) = x³ + x² – 2 với mọi x

b) ( z² – z + 1)( az² + bz + c) = 2z⁴ – z³ + 2z² + 1 với mọi Z

Answer 1

a: \(\Leftrightarrow a\cdot x^3+b\cdot x^2+ac\cdot x^2+b\cdot cx+2ax+2b=x^3+x^2-2\)

\(\Leftrightarrow\left\{{}\begin{matrix}a=1\\b+ac=1\\bc+2a=0\\2b=-2\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}a=1\\b=-1\\c=2\\-1\cdot2+2\cdot1=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}a=1\\b=-1\\c=2\end{matrix}\right.\)

b: \(\left(z^2-z+1\right)\left(az^2+bz+c\right)\)

\(=az^4+bz^3+cz^2-az^3-bz^2-cz+az^2+bz+c\)

\(=az^4+z^3\left(b-a\right)+z^2\left(c-b+a\right)+z\left(-c+b\right)+c\)

Theo đề, ta có: a=2; \(\left\{{}\begin{matrix}b-a=-1\\c-b+a=2\\-c+b=0\\c=1\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}b=-1+a=-1+2=1\\c=2+b-a=2+1-2=1\\1-1=0\\c=1\end{matrix}\right.\)

=>a=2; b=1; c=1