利用平方差公式作因式分解

若一多項式可寫為 $a^2-b^2$ 的形式,可利用平方差公式 $a^2-b^2=(a+b)(a-b)$ 進行因式分解。

因式分解下列各式:

  1. $x^2-81$
  2. $8-2x^2$
  3. $(2x+1)^2-(x+2)^2$
  4. $(x-3)^2-1$

  1. $\begin{array} {rl} & x^2-81 \\ = & (x)^2-(9)^2 \\ = & (x+9)(x-9) \end{array}$
  2. $\begin{array} {rl} & 8-2x^2 \\ = & 2(4-x^2) \\ = & 2(2^2-x^2) \\ = & 2(2+x)(2-x) \end{array}$
  3. $\begin{array} {rl} & (2x+1)^2-(x+2)^2 \\ = & [(2x+1)+(x+2)][(2x+1)-(x+2)] \\ = & (2x+1+x+2)(2x+1-x-2) \\ = & (3x+3)(x-1) \\ = & 3(x+1)(x-1) \end{array}$
  4. $\begin{array} {rl} & (x-3)^2-1 \\ = & (x-3)^2-1^2 \\ = & [(x-3)+1][(x-3)-1] \\ = & (x-3+1)(x-3-1) \\ = & (x-2)(x-4) \end{array}$

觀念影片