因式分解指的是將一個多項式化為一個連乘的因式。
常用的因式分解方法:提公因式法、公式法、十字相乘法、分組分解法。
將一個多項式中的公有部分提出來,化為連乘的形式。
例:4x2y3+8xy=4xy(x+2y2){\displaystyle 4x^{2}y^{3}+8xy=4xy(x+2y^{2})}
x2−y2=(x+y)(x−y){\displaystyle x^{2}-y^{2}=(x+y)(x-y)}
(x+y)2=x2+2xy+y2{\displaystyle (x+y)^{2}=x^{2}+2xy+y^{2}} (x+y)3=x3+3x2y+3xy2+y3{\displaystyle (x+y)^{3}=x^{3}+3x^{2}y+3xy^{2}+y^{3}}
(x−y)2=x2−2xy+y2{\displaystyle (x-y)^{2}=x^{2}-2xy+y^{2}}
將x2+(p+q)x+pq{\displaystyle x^{2}+(p+q)x+pq} 的形式分解成(x+p)(x+q){\displaystyle (x+p)(x+q)} 的形式
証明如下:
x2+(p+q)x+pq{\displaystyle x^{2}+(p+q)x+pq} =x(x+p)+q(x+p){\displaystyle =x(x+p)+q(x+p)} =(x+p)(x+q){\displaystyle =(x+p)(x+q)}
將多項式適當分組,然後再分解。
例: 2x3+4x2+x+2{\displaystyle 2x^{3}+4x^{2}+x+2} =2x2(x+2)+x+2{\displaystyle =2x^{2}(x+2)+x+2} =(2x2+1)(x+2){\displaystyle =(2x^{2}+1)(x+2)}
9x2+25x−44{\displaystyle 9x^{2}+25x-44} =9x2+36x−11x−44{\displaystyle =9x^{2}+36x-11x-44} (25拆成36–11)=9x(x+4)−11(x+4){\displaystyle =9x(x+4)-11(x+4)}
=(9x−11)(x+4){\displaystyle =(9x-11)(x+4)}