Cho hai đa thức P(x)=-6(x^5)-4(x^4)+3(x^2)-2x; Q(x)=2(x^5)-4(x^4)-2(x^3)+2(x^2)-x-3.

* Hướng dẫn giải

Đáp án cần chọn là C.

Ta có:

$$\begin{gathered} 2Q\left(x\right)=2\left(2x^5-4x^4-2x^3+2x^2-x-3\right) \\ =4x^5-8x^4-4x^3+4x^2-2x-3 \end{gathered}$$

Khi đó:

$$\begin{gathered} P\left(x\right)-2Q\left(x\right) \\ =-6x^5-4x^4+3x^2-2x-\left(4x^5-8x^4-4x^3+4x^2-2x-6\right) \\ =-6x^5-4x^4+3x^2-2x-4x^5+8x^4+4x^3-4x^2+2x+6 \\ =\left(-6x^5-4x^5\right)+\left(-4x^4+8x^4\right)+4x^3+\left(3x^2-4x^2\right)+\left(-2x+2x\right)+6 \\ =-10x^5+4x^4+4x^3-x^2+6 \end{gathered}$$

Nên

 $$\begin{gathered} P\left(x\right)-2Q\left(x\right)=N\left(x\right)-x^2+6 \\ \Rightarrow N=P\left(x\right)-2Q\left(x\right)-\left(-x^2+6\right) \\ =-10x^5+4x^4+4x^3-x^2+6-\left(-x^2+6\right) \\ =-10x^5+4x^4+4x^3-x^2+6+x^2-6 \\ =-10x^5+4x^4+4x^3 \end{gathered}$$

Nên $N\left(x\right)=-10x^5+4x^4+4x^3$.