Phân tích các đa thức sau thành nhân tử:

$$\begin{gathered} a) x^3 + 2x^{2}y + x{y}^2 – 9x; \\ b) 2x – 2y – x^2 + 2xy – y^2; \\ c) x^4 – 2x^2; \end{gathered}$$

Giải:

$$\begin{gathered} a) x^3 + 2x^{2}y + x{y}^2 – 9x \\ =x(x^2 +2xy + y^2 – 9) \\ = x[(x^2 + 2xy + y^2) – 9] \\ = x[{(x + y)}^2 – 3^2] \\ = x(x + y – 3\left)\right(x + y + 3) \end{gathered}$$

$$\begin{gathered} b) 2x – 2y – x^2 + 2xy – y^2 \\ = (2x – 2y) – (x^2 – 2xy + y^2) \\ = 2(x – y) – {(x – y)}^2= (x – y\left)\right[2 – (x – y)] \\ = (x – y\left)\right(2 – x + y) \end{gathered}$$

$$\begin{gathered} c) x^4 – 2x^2 \\ = x^2(x^2 – {\left(\sqrt{2}\right)}^2) \\ = x^2(x - \sqrt{2}\left)\right(x + \sqrt{2}). \end{gathered}$$