Phân tích các đa thức sau thành nhân tử:

$$\begin{gathered} a)x^2+4x-y^2+4; \\ b)3x^2+6xy+3y^2-3z^2; \\ c)x^2-2xy+y^2-z^2+2zt-t^2; \end{gathered}$$

Giải:

$$\begin{gathered} a. x^2+ 4x – y^2 + 4 \\ = (x^2 + 4x + 4) - y^2 \\ = {(x + 2)}^2 – y^2 \\ = (x + 2 – y)(x + 2 + y) \end{gathered}$$

$$\begin{gathered} b. 3x^2 + 6xy + 3y^2 – 3z^{2^{}} \\ = 3\left[\right(x^2 + 2xy + y^2) – z^2] \\ = 3[{(x + y)}^2– z^2] \\ = 3(x + y – z)(x + y + z) \end{gathered}$$

$$\begin{gathered} c. x^2 – 2xy + y^2– z^2 + 2zt – t^2 \\ = (x^2 – 2xy + y^2) – (z^2– 2zt + t^2) \\ = {(x – y)}^2– {(z – t)}^2= \left[\right(x – y) – (z – t\left)\right] . \left[\right(x – y) + (z – t\left)\right] \\ = (x – y – z + t)(x – y + z – t) \end{gathered}$$