Làm tính nhân:

$a) \left(x^2-2x+1\right)\left(x-1\right)$

$b) \left(x^3-2x^2+x-1\right)\left(5-x\right)$

Từ câu b), hãy suy ra kết quả phép nhân: $\left(x^3-2x^2+x-1\right)\left(x-5\right)$

Giải:

$$\begin{gathered} a) \left(x^2-2x+1\right)\left(x-1\right) \\ =x^2.x+x^2.\left(-1\right)+\left(-2x\right).x+\left(-2x\right).\left(-1\right)+1.x+1.\left(-1\right) \\ =x^3-x^2-2x^2+2x+x-1 \\ =x^3-3x^2+3x-1 \end{gathered}$$

$$\begin{gathered} b) \left(x^3-2x^2+x-1\right)\left(5-x\right) \\ =x^3.5+x^3.\left(-x\right)+\left(-2x^2\right).5+\left(-2x^2\right).\left(-x\right)+x.5+x.\left(-x\right)+\left(-1\right).5+\left(-1\right).\left(-x\right) \\ =5x^3-x^4-10x^2+2x^3+5x-x^2-5+x \\ =-x^4+7x^3-11x^2+6x-5 \end{gathered}$$

Suy ra kết quả của phép nhân:

$$\begin{gathered} \left(x^3-2x^2+x-1\right)\left(x-5\right) \\ =\left(x^3-2x^2+x-1\right)\left(-\left(5-x\right)\right) \\ =-\left(x^3-2x^2+x-1\right)\left(5-x\right) \\ =-\left(-x^4+7x^3-11x^2+6x-5\right) \\ =x^4-7x^3+11x^2-6x+5 \end{gathered}$$