分解因式:$x^4+4$
【答案】$\left( {x}^{2}+2x+2 \right)\left( {x}^{2}-2x+2 \right)$
- 原式$={x}^{4}+4$
$\hspace{2em}={x}^{4}+4{x}^{2}+4-4{x}^{2}$
$\hspace{2em}={\left({x}^{2}+2\right)}^{2}-{\left(2x\right)}^{2}$
$\hspace{2em}=\left( {x}^{2}+2x+2 \right)\left( {x}^{2}-2x+2 \right)$.
分解因式:${x}^{4}+{2}^{6}$.
【答案】$\left( {x}^{2}+4x+8 \right)\left( {x}^{2}-4x+8 \right)$
- 原式$=x^4+16x^2+64-16x^2$
$\hspace{2em}=\left(x^2+8\right)^2-\left(4x\right)^2$
$\hspace{2em}=\left( {x}^{2}+4x+8 \right)\left( {x}^{2}-4x+8 \right)$
分解因式:${x}^{2}-{y}^{2}+2x+6y-8$.
【答案】$\left( x+y-2 \right)\left( x-y+4 \right)$
- 原式$=x^2+2x+1-\left(y^2-6y+9\right)$
$\hspace{2em}=\left(x+1\right)^2-\left(y-3\right)^2$
$\hspace{2em}=\left( x+y-2 \right)\left( x-y+4 \right)$
分解因式:${x}^{4}-4x+3$.
【答案】${\left( x-1 \right)}^{2}\left( {x}^{2}+2x+3 \right)$
- 原式$={x}^{4}-x-3x+3$
$\hspace{2em}=\left( {x}^{4}-x \right)-\left( 3x-3 \right)$
$\hspace{2em}=x\left( x-1 \right)\left( {x}^{2}+x+1 \right)-3\left( x-1 \right)$
$\hspace{2em}=\left( x-1 \right)\left( {x}^{3}+{x}^{2}+x-3 \right)$.
$\hspace{2em}={\left( x-1 \right)}^{2}\left( {x}^{2}+2x+3 \right)$
分解因式:${x}^{4}-3{x}^{2}+1$.
【答案】$\left( {x}^{2}+x-1 \right)\left( {x}^{2}-x-1 \right)$
- 原式$=x^4-2x^2+1-x^2$
$\hspace{2em}=\left(x^2-1\right)^2-x^2$
$\hspace{2em}=\left( {x}^{2}+x-1 \right)\left( {x}^{2}-x-1 \right)$
分解因式:${x}^{4}-23{x}^{2}+1$.
【答案】$\left( {x}^{2}+5x+1 \right)\left( {x}^{2}-5x+1 \right)$
- 原式$=x^4+2x^2+1-25x^2$
$\hspace{2em}=\left(x^2+1\right)^2-\left(5x\right)^2$
$\hspace{2em}=\left( {x}^{2}+5x+1 \right)\left( {x}^{2}-5x+1 \right)$
分解因式:${x}^{4}-47{x}^{2}+1$.
【答案】$\left( {x}^{2}+7x+1 \right)\left( {x}^{2}-7x+1 \right)$
- 原式$=x^4+2x^2+1-49x^2$
$\hspace{2em}=\left(x^2+1\right)^2-\left(7x\right)^2$
$\hspace{2em}=\left( {x}^{2}+7x+1 \right)\left( {x}^{2}-7x+1 \right)$
分解因式:${x}^{8}+{x}^{4}+1$.
【答案】$\left( {x}^{2}-x+1 \right)\left( {x}^{2}+x+1 \right)\left( {x}^{4}-{x}^{2}+1 \right)$
- 原式$=x^8+2x^4+1-x^4$
$\hspace{2em}=\left(x^4+1\right)^2-\left(x^2\right)^2$
$\hspace{2em}=\left(x^4+x^2+1\right)\left(x^4-x^2+1\right)$
$\hspace{2em}=\left(x^4+2x^2+1-x^2\right)\left(x^4-x^2+1\right)$
$\hspace{2em}=\left[\left(x^2+1\right)^2-x^2\right]\left(x^4-x^2+1\right)$
$\hspace{2em}=\left( {x}^{2}-x+1 \right)\left( {x}^{2}+x+1 \right)\left( {x}^{4}-{x}^{2}+1 \right)$
分解因式:${x}^{12}-3{x}^{6}+1$.
【答案】$\left( {x}^{6}+{x}^{3}-1 \right)\left( {x}^{6}-{x}^{3}-1 \right)$
- 原式$=x^{12}-2x^6+1-x^6$
$\hspace{2em}=\left(x^6-1\right)^2-\left(x^3\right)^2$
$\hspace{2em}=\left( {x}^{6}+{x}^{3}-1 \right)\left( {x}^{6}-{x}^{3}-1 \right)$
分解因式:$-14{x}^{2}{y}^{2}+{x}^{4}+{y}^{4}$.
【答案】$\left(x^2+4xy+y^2\right)\left(x^2-4xy+y^2\right)$
- 原式$=x^4+2x^2y^2+y^4-16x^2y^2$
$\hspace{2em}=\left(x^2+y^2\right)^2-\left(4xy\right)^2$
$\hspace{2em}=\left(x^2+4xy+y^2\right)\left(x^2-4xy+y^2\right)$
分解因式:${x}^{4}-7{x}^{2}{y}^{2}+81{y}^{4}$.
【答案】$\left( {x}^{2}+5xy+9{y}^{2} \right)\left( {x}^{2}-5xy+9{y}^{2} \right)$
- 原式$=x^4+18x^2y^2+81y^4-25x^2y^2$
$\hspace{2em}=\left(x^2+9y^2\right)^2-\left(5xy\right)^2$
$\hspace{2em}=\left( {x}^{2}+5xy+9{y}^{2} \right)\left( {x}^{2}-5xy+9{y}^{2} \right)$
分解因式:${a}^{4}+{a}^{2}{b}^{2}+{b}^{4}$.
【答案】$\left( {a}^{2}+ab+{b}^{2} \right)\left( {a}^{2}-ab+{b}^{2} \right)$
- 原式$=\left( {a}^{4}+2{a}^{2}{b}^{2}+{b}^{4} \right)-{a}^{2}{b}^{2}$
$\hspace{2em}={\left( {a}^{2}+{b}^{2} \right)}^{2}-{\left( ab \right)}^{2}$
$\hspace{2em}=\left( {a}^{2}+ab+{b}^{2} \right)\left( {a}^{2}-ab+{b}^{2} \right)$.
分解因式:${a}^{3}+3{a}^{2}+3a+2$.
【答案】$\left( a+2 \right)\left( {a}^{2}+a+1 \right)$.
- 原式$=\left( {a}^{3}+3{a}^{2}+3a+1 \right)+1$
$\hspace{2em}={\left( a+1 \right)}^{3}+{1}^{3}$
$\hspace{2em}=\left( a+1+1 \right)\left[ {\left( a+1 \right)}^{2}-\left( a+1 \right)+1 \right]$
$\hspace{2em}=\left( a+2 \right)\left( {a}^{2}+a+1 \right)$.
分解因式:${a}^{3}+3{a}^{2}+3a+{b}^{3}+3{b}^{2}+3b+2$.
【答案】$=\left( a+b+2 \right)\left( {a}^{2}-ab+{b}^{2}+a+b+1 \right)$
- 原式$=\left( {a}^{3}+3{a}^{2}+3a+1 \right)+\left( {b}^{3}+3{b}^{2}+3b+1 \right)$
$\hspace{2em}={\left( a+1 \right)}^{3}+{\left( b+1 \right)}^{3}$
$\hspace{2em}=\left( a+b+2 \right)\left[ {\left( a+1 \right)}^{2}-\left( a-1 \right)\left( b+1 \right)+{\left( b+1 \right)}^{2} \right]$
$\hspace{2em}=\left( a+b+2 \right)\left( {a}^{2}-ab+{b}^{2}+a+b+1 \right)$.
分解因式:${\left( x+1 \right)}^{4}+{\left( {x}^{2}-1 \right)}^{2}+{\left( x-1 \right)}^{4}$.
【答案】$\left( 3{x}^{2}+1 \right)\left( {x}^{2}+3 \right)$
- 原式$={\left( x+1 \right)}^{4}+2{\left( {x}^{2}-1 \right)}^{2}-{\left( {x}^{2}-1 \right)}^{2}+{\left( x-1 \right)}^{4}$
$\hspace{2em}=\left[{\left( x+1 \right)}^{4}+2{\left( x+1 \right)}^{2}{\left( x-1 \right)}^{2}+{\left( x-1 \right)}^{4}\right]-{\left( {x}^{2}-1 \right)}^{2}$
$\hspace{2em}={\left[{\left( x+1 \right)}^{2}+{\left( x-1 \right)}^{2}\right]}^{2}-{\left( {x}^{2}-1 \right)}^{2}$
$\hspace{2em}={\left( 2{x}^{2}+2 \right)}^{2}-{\left( {x}^{2}-1 \right)}^{2}=\left( 3{x}^{2}+1 \right)\left( {x}^{2}+3 \right)$
因式分解:${x}^{3}+3{x}^{2}+5x+6$.
【答案】$\left( x+2 \right)\left( {x}^{2}+x+3 \right)$.
- 原式$=\left( {x}^{3}+2{x}^{2} \right)+\left( {x}^{2}+5x+6 \right)$,
$\hspace{2em}={x}^{2}\left( x+2 \right)+\left( x+2 \right)\left( x+3 \right)$,
$\hspace{2em}=\left( x+2 \right)\left( {x}^{2}+x+3 \right)$.
因式分解:$9x^4+5x^2+1$.
【答案】$\left(3x^{2}+1+x\right)\left(3x^{2}+1-x\right)$.
- 原式$=9x^{4}+5x^{2}+1+x^{2}-x^{2}$
$\hspace{2em}=\left(3x^2+1\right)-x^2$
$\hspace{2em}=\left(3x^{2}+1+x\right)\left(3x^{2}+1-x\right)$.
分解因式:$9{x}^{2}-6x-{y}^{2}+4y-3$
【答案】$\left(3x+y-3\right)\left(3x-y+1\right)$
- 原式$=\left(9{x}^{2}-6x+1\right)-\left({y}^{2}-4y+4\right)$
$\hspace{2em}={\left(3x-1\right)}^{2}-{\left(y-2\right)}^{2}$
$\hspace{2em}=\left(3x+y-3\right)\left(3x-y+1\right)$.
分解因式:$4{x}^{3}-31x+15$.
【答案】$\left(2x-1\right)\left(2x-5\right)\left(x+3\right)$.
- 原式$=4{x}^{3}-x-30x+15$
$\hspace{2em}=x\left(4{x}^{2}-1\right)-15\left(2x-1\right)$
$\hspace{2em}=x\left(2x+1\right)\left(2x-1\right)-15\left(2x-1\right)$
$\hspace{2em}=\left(2x-1\right)\left[ x\left(2x+1\right)-15 \right]$
$\hspace{2em}=\left(2x-1\right)\left(2{x}^{2}+x-15\right)$
$\hspace{2em}=\left(2x-1\right)\left(2x-5\right)\left(x+3\right)$.
计算:$y\left(x-y+10\right)-2\left(x+8\right)$.
【答案】$\left(y-2\right)\left(x+8-y\right)$.
- 原式$=y\left(x+8+2-y\right)-2\left(x+8\right)$
$\hspace{2em}=y\left(x+8\right)+y\left(2-y\right)-2\left(x+8\right)$
$\hspace{2em}=\left(x+8\right)\left(y-2\right)-y\left(y-2\right)$
$\hspace{2em}=\left(y-2\right)\left(x+8-y\right)$.
分解因式:${x}^{4}+{x}^{3}-3{x}^{2}-4x-4$.
【答案】$\left(x+2\right)\left(x-2\right)\left({x}^{2}+x+1\right)$
- 原式$={x}^{4}+{x}^{3}+{x}^{2}-4{x}^{2}-4x-4$
$\hspace{2em}=x^2\left(x^2+x+1\right)-4\left(x^2+x+1\right)$
$\hspace{2em}=\left(x^2-4\right)\left(x^2+x+1\right)$
$\hspace{2em}=\left(x+2\right)\left(x-2\right)\left(x^2+x+1\right)$
分解因式:${a}^{2}-4ab+3{b}^{2}+2bc-{c}^{2}$.
【答案】$\left(a-b-c\right)\left(a-3b+c\right)$.
- 原式$={a}^{2}-4ab+4{b}^{2}-{b}^{2}+2bc-{c}^{2}$
$\hspace{2em}={\left(2-2b\right)}^{2}-{\left(b-c\right)}^{2}$
$\hspace{2em}=\left(a-2b+b-c\right)\left(a-2b-b+c\right)$
$\hspace{2em}=\left(a-b-c\right)\left(a-3b+c\right)$.
因式分解:${m}^{4}+4{n}^{4}$.
【答案】$\left({m}^{2}+2{n}^{2}+2mn\right)\left({m}^{2}+2{n}^{2}-2mn\right) $.
- 原式$={m}^{4}+4{m}^{2}{n}^{2}+4{n}^{4}-4{m}^{2}{n}^{2}$
$\hspace{2em}={\left({m}^{2}+2{n}^{2}\right)}^{2}-{\left(2mn\right)}^{2}$
$\hspace{2em}=\left({m}^{2}+2{n}^{2}+2mn\right)\left({m}^{2}+2{n}^{2}-2mn\right)$.
分解因式:$2{x}^{3}-5{x}^{2}+5x-3$.
【答案】$\left( 2x-3 \right)\left( {x}^{2}-x+1 \right)$.
- 原式$=2{x}^{3}-3{x}^{2}-2{x}^{2}+3x+2x-3$,
$\hspace{2em}={x}^{2}\left( 2x-3 \right)-x\left( 2x-3 \right)+\left( 2x-3 \right)$,
$\hspace{2em}=\left( 2x-3 \right)\left( {x}^{2}-x+1 \right)$.
分解因式:$-{a}^{4}-{b}^{4}-{c}^{4}+2{a}^{2}{b}^{2}+2{b}^{2}{c}^{2}+2{c}^{2}{a}^{2}$.
【答案】
- 原式$=-\left( {a}^{4}+{b}^{4}+{c}^{4}-2{a}^{2}{b}^{2}-2{b}^{2}{c}^{2}-2{c}^{2}{a}^{2} \right)$
$\hspace{2em}=-\left( {a}^{4}+{b}^{4}+{c}^{4}+2{a}^{2}{b}^{2}-2{b}^{2}{c}^{2}-2{c}^{2}{a}^{2}-4{a}^{2}{b}^{2} \right)$
$\hspace{2em}=-\left[ {\left( {a}^{2}+{b}^{2}-{c}^{2} \right)}^{2}-{\left( 2ab \right)}^{2} \right]$
$\hspace{2em}=-\left( {a}^{2}+{b}^{2}-{c}^{2}+2ab \right)\left( {a}^{2}+{b}^{2}-{c}^{2}-2ab \right)$
$\hspace{2em}=-\left[ {\left( a+b \right)}^{2}-{c}^{2} \right]\left[ {\left( a-b \right)}^{2}-{c}^{2} \right]$
$\hspace{2em}=-\left( a+b+c \right)\left( a+b-c \right)\left( a-b+c \right)\left( a-b-c \right)$
$\hspace{2em}=\left( a+b+c \right)\left( a+b-c \right)\left( a-b+c \right)\left( b+c-a \right)$.
因式分解:${a}^{3}+{b}^{3}+{c}^{3}-3abc$
【答案】$\left(a+b+c\right)\left({a}^{2}+{b}^{2}+{c}^{2}-ab-bc-ca\right)$
- 原式$={a}^{3}+{b}^{3}+3{a}^{2}b+3a{b}^{2}+{c}^{3}-3{a}^{2}b-3a{b}^{2}-3abc$
$\hspace{2em}={\left(a+b\right)}^{3}+{c}^{3}-3ab\left(a+b+c\right)$
$\hspace{2em}=\left(a+b+c\right)\left({a}^{2}+{b}^{2}+2ab+{c}^{2}-ac-bc\right)-3ab\left(a+b+c\right)$
$\hspace{2em}=\left(a+b+c\right)\left({a}^{2}+{b}^{2}+{c}^{2}-ab-bc-ca\right)$.