Question

Cho \(P\left(x\right)=x^4+ax^3-bx^2+cx+d\)

\(P\left(-2\right)=6;P\left(-4\right)=12;P\left(-6\right)=18\)

Tính \(\dfrac{P\left(0\right)+P\left(-8\right)+437P\left(-2\right)}{2020}\)

Answer 1

Đặt \(f\left(x\right)=P\left(x\right)+3x\)

\(f\left(x\right)=P\left(x\right)+3x\\ \Leftrightarrow\left\{{}\begin{matrix}f\left(-2\right)=0\\f\left(-4\right)=0\\f\left(-6\right)=0\end{matrix}\right.\Leftrightarrow f\left(x\right)=\left(x-m\right)\left(x+2\right)\left(x+4\right)\left(x+6\right)\\ \Leftrightarrow P\left(x\right)=\left(x-m\right)\left(x+2\right)\left(x+4\right)\left(x+6\right)+3x\\ \Leftrightarrow\left\{{}\begin{matrix}P\left(-2\right)=0\\P\left(0\right)=-m\cdot2\cdot4\cdot6+0=-48m\\P\left(-8\right)=\left(-8-m\right)\left(-6\right)\left(-4\right)\left(-2\right)-24=48m+360\end{matrix}\right.\)

Do đó \(A=\dfrac{-48m+48m+360+0}{2020}=\dfrac{360}{2020}=\dfrac{18}{101}\)