Question

Tìm a, b biết :

a, \(x^4+ax^2+b⋮x^2-x+1\)

b, \(ax^3+bx^2+5x-50⋮\left(x^2+3x-10\right)\)

c, \(ax^4+bx^3+1⋮\left(x-1\right)^2\)

d, \(x^4+4⋮\left(x^2+ax+b\right)\)

Answer 1

b, \(ax^3+bx^2+5x-50⋮\left(x^2+3x-10\right)\)

\(\Rightarrow f\left(x\right)=ax^3+bx^2+5x-50⋮\left(x-2\right)\left(x+5\right)\)\(\Leftrightarrow\left\{{}\begin{matrix}f\left(2\right)=0\\f\left(-5\right)=0\end{matrix}\right.\)

\(\Rightarrow\left\{{}\begin{matrix}f\left(2\right)=8a+4b+10-50=0\\f\left(-5\right)=-125a+25b-25-50=0\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}f\left(2\right)=4\left(2a+b\right)=40\\f\left(-5\right)=-25\left(5a-b\right)=75\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}f\left(2\right)=2a+b=1\\f\left(-5\right)=5a-b=-3\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}a=-\dfrac{2}{7}\\b=\dfrac{11}{7}\end{matrix}\right.\)