式の展開、因数分解 - CANADA'S WINDVIEW

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式の展開、因数分解に関する公式です。

基本公式

$\displaystyle ( a + b ) ^ 2 = a ^ 2 + 2 a b + b ^ 2$

$\displaystyle ( a - b ) ^ 2 = a ^ 2 - 2 a b + b ^ 2$

$\displaystyle a ^ 2 - b ^ 2 = ( a - b ) ( a + b )$

$\displaystyle ( a + b ) ^ 3 = a ^ 3 + 3 a ^ {2} b + 3 a b ^ {2} + b ^ 3 = a^3 + b^3 + 3 a b ( a + b )$

$\displaystyle ( a - b ) ^ 3 = a ^ 3 - 3 a ^ {2} b + 3 a b ^ {2} - b ^ 3 = a^3 - b^3 - 3 a b ( a - b )$

$\displaystyle ( x + a ) ( x + b ) = x ^ 2 + ( a + b ) x + a b$

$\displaystyle ( a x + b y ) ( c x + d y ) = a c x ^ 2 + ( b c + a d ) x y + b d y ^ 2$

発展公式

$\displaystyle a ^ 2 + b ^ 2 = ( a + b ) ^ 2 - 2 a b$

$\displaystyle a ^ 2 + b ^ 2 = ( a - b ) ^ 2 + 2 a b$

$\displaystyle a ^ 2 + b ^ 2 + c ^ 2 = ( a + b + c ) ^ 2 - 2 ( a b + b c + c a )$

$\displaystyle a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = ( a + b + c + d ) ^ 2 - 2 ( a b + b c + c d + d a + a c + b d )$

$\displaystyle a b ( a - b ) + b c ( b - c ) + c a ( c - a ) = - ( a - b ) ( b - c ) ( c - a )$

$\displaystyle a ^ 3 - b ^ 3 = ( a - b ) \left( a ^ 2 + a b + b ^ 2 \right)$

$\displaystyle a ^ 4 - b ^ 4 = ( a - b ) ( a + b ) \left( a ^ 2 + b ^ 2 \right)$

$\displaystyle a ^ 3 + b ^ 3 = ( a + b ) \left( a ^ 2 - a b + b ^ 2 \right)$

$\displaystyle a ^ 5 + b ^ 5 = ( a + b ) \left( a ^ 4 - a ^ 3 b +a ^ 2 b ^ 2 - a b ^ 3 + b ^ 4 \right)$

$\displaystyle a ^ 3 + b ^ 3 + c ^ 3 - 3 a b c = ( a + b )^3 - 3 a b ( a + b ) -3 a b c + c^3 \\ = ( a + b )^3 + c^3 - 3 a b ( a + b + c ) = \left\{ ( a + b ) + c \right\}^3 - 3 ( a + b ) c \left\{ ( a + b ) + c \right\} - 3 a b ( a + b + c ) \\ = ( a + b + c ) \left\{ ( a + b + c )^2 - 3 ( a + b ) c - 3 a b \right\} = ( a + b + c ) \left\{ ( a + b + c )^2 - 3 ( a b + b c + c a ) \right\} \\ = ( a + b + c ) \left\{ a^2 + b^2 + c^2 + 2 ( a b + b c + c a ) - 3 ( a b + b c + c a ) \right\} \\ = ( a + b + c ) \left( a ^ 2 + b ^ 2 + c ^ 2 - a b - b c - c a \right)$

$\displaystyle a ^ 3 + b ^ 3 + c ^ 3 + d ^ 3 - 3 ( a b c + b c d + c d a + d a b ) \\ \displaystyle = ( a + b + c + d ) \left( a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 - a b - b c - c d - d a - a c - b d \right)$

nが奇数の時

$\displaystyle a ^ n - b ^ n = ( a - b ) \left( a ^ {n - 1} + a ^ {n - 2} b + a ^ {n - 3} b ^ 2 + \ldots + a b ^ {n - 2} + b ^ {n - 1} \right)$

例

$a^9 - b^9 = ( a - b ) \left( a^8 + a^7 b + a^6 b^2 + a^5 b^3 + a^4 b^4 + a^3 b^5 + a^2 b^6 + a b^7 + b^8 \right)$

興味深い因数分解の例

$\displaystyle a ^ 2 + b ^ 2 + c ^ 2 - a b - b c - c a \\ \displaystyle = \frac{1}{2} \left( 2 a ^ 2 + 2 b ^ 2 + 2 c ^ 2 - 2 a b - 2 b c - 2 c a \right) \\ \displaystyle = \frac{1}{2} \left( a ^ 2 - 2 a b + b ^ 2 + b ^ 2 - 2 b c + c ^ 2 + c ^ 2 - 2 c a + a ^ 2 \right) \\ \displaystyle = \frac{1}{2} \left\{ ( a - b ) ^ 2 + ( b - c ) ^ 2 + ( c - a ) ^ 2 \right\}$

$a^4 + a^2 b^2 + b^4 = \left( a^2 \right)^2 + 2 \left( a^2 \right) \left( b^2 \right)+ \left( b^2 \right)^2 - \left( a^2 \right) \left( b^2 \right) = \left( a^2 + b^2 \right)^2 - ( a b )^2 \\ = \left\{ \left( a^2 + b^2 \right) + a b \right\} \left\{ \left( a^2 + b^2 \right) - a b \right\} = \left( a^2 + a b + b^2 \right) \left( a^2 - a b + b^2 \right)$

特に $b = 1$ の時、

$a^4 + a^2 + 1 = \left( a^2 + a + 1 \right) \left( a^2 -a + 1 \right)$

$a^4 + 4 = \left( a^2 \right)^2 + 4 a^2 + 4 - 4 a^2 = \left( a^2 + 2 \right)^2 - ( 2 a )^2 = \left\{ \left( a^2 + 2 \right) + 2 a \right\} \left\{ \left( a^2 + 2 \right) - 2 a \right\} \\ = \left( a^2 + 2 a + 2 \right) \left( a^2 - 2 a + 2 \right)$

$( x + 1 ) x ( x - 1 ) ( x - 2 ) + 1 = \left\{ x ( x - 1 ) \right\} \left\{ ( x + 1 ) ( x - 2 ) \right\} + 1 = \left( x^2 - x \right) \left( x^2 - x - 2 \right) + 1 \\ = \left( x^2 - x \right) \left\{ \left( x^2 - x \right) - 2 \right\} + 1 = \left( x^2 - x \right)^2 - 2 \left( x^2 - x \right) + 1 = \left\{ \left( x^2 - x \right) - 1 \right\}^2 \\ = \left( x^2 - x - 1 \right)^2$

$a^5 - a^4 - 1 = a^5 - a^4 + a^3 - a^3 - 1 = a^3 \left( a^2 - a + 1 \right) - \left( a^3 + 1 \right) \\ \displaystyle = a^3 \left( a^2 - a + 1 \right) - ( a + 1 ) \left( a^2 - a + 1 \right) = \left( a^3 - a - 1 \right) \left( a^2 - a + 1 \right)$

$\left( x + y + z \right)^3 - \left( x^3 + y^3 + z^3 \right) = \left\{ \left( x + y + z \right)^3 - x^3 - \left( y^3 + z^3 \right) \right\} \\ = \left\{ ( x + y + z ) - x \right\} \left\{ ( x + y + z )^2 + ( x + y + z ) x + x^2 \right\} - ( y + z ) \left( y^2 - y z + z^2 \right) \\ = ( y + z ) \left\{ ( x + y + z )^2 + ( x + y + z ) x + x^2 - \left( y^2 - y z + z^2 \right) \right\} \\ = ( y + z ) \left( x^2 + y^2 + z^2 + 2 x y + 2 y z + 2 z x + x^2 + x y + z x - y^2 + y z - z^2 \right) \\ = ( y + z ) \left( 3 x^2 + 3 x y + 3 y z + 3 z x \right) = 3 ( y + z ) \left( x^2 + x y + y z + z x \right) \\ = 3 ( y + z ) \left\{ x^2 + x ( y + z ) + y z \right\} = 3 ( y + z ) ( x + y ) ( x + z ) = 3 ( x + y ) ( y + z ) ( x + z )$

$\displaystyle x^4 - 2 x^3 + 3 x^2 - 2 x + 1 = x^2 \left( x^2 - 2 x + 3 - 2 \cdot \frac{ 1 }{ x } + \frac{ 1 }{ x^2 } \right) = x^2 \left\{ \left( x^2 + \frac{ 1 }{ x^2 } \right) - 2 \left( x + \frac{ 1 }{ x } \right) + 3 \right\} \\ \displaystyle = x^2 \left\{ \left( x + \frac{ 1 }{ x } \right)^2 - 2 - 2 \left( x + \frac{ 1 }{ x } \right) + 3 \right\} = x^2 \left\{ \left( x + \frac{ 1 }{ x } \right)^2 - 2 \left( x + \frac{ 1 }{ x } \right) + 1 \right\} \\ \displaystyle = x^2 \left( x + \frac{ 1 }{ x } - 1 \right)^2 = \left( x^2 - x + 1 \right)^2$

$( a + b + c )^3 - \left( a^3 + b^3 + c^3 \right)$

演習問題

$a^{18} - b^{18} = \left( a^{9} \right)^2 - \left( b^{9} \right)^2 = \left( a^{9} + b^{9} \right) \left( a^{9} - b^{9} \right) = \left\{ \left( a^{3} \right)^3 + \left( b^{3} \right)^3 \right\} \left\{ \left( a^{3} \right)^3 - \left( b^{3} \right)^3 \right\} \\ = \left( a^3 + b^3 \right) \left( a^6 - a^3 b^3 + b^6 \right) \left( a^3 - b^3 \right) \left( a^6 + a^3 b^3 + b^6 \right) \\ = ( a + b ) ( a - b ) \left( a^2 - a b + b^2 \right) \left( a^2 + a b + b^2 \right) \left( a^6 - a^3 b^3 + b^6 \right) \left( a^6 + a^3 b^3 + b^6 \right)$

特に $b = 1$ の時、

$a^{18} - 1 = ( a + 1 )( a - 1 ) \left( a^2 - a + 1 \right) \left( a^2 + a + 1 \right) \left( a^6 - a^3 + 1 \right) \left( a^6 + a^3 + 1 \right)$

$x^4 - 5 x^2 + 4 = \left( x^2 \right)^2 - 5 \left( x^2 \right) + 4 = \left( x^2 - 1 \right) \left( x^2 - 4 \right) = \left( x + 1 \right) \left( x - 1 \right) \left( x + 2 \right) \left( x - 2 \right)$

$x^4 + x^2 - 20 = \left( x^2 \right)^2 + \left( x^2 \right) - 20 = \left( x^2 - 4 \right) \left( x^2 + 5 \right) = ( x + 2 ) ( x - 2 ) \left( x^2 + 5 \right)$

$x^4 + x^2 + 25 = x^4 + 10 x^2 + 25 - 10 x^2 + x^2 = \left( x^2 + 5 \right)^2 - 9 x^2 = \left( x^2 + 5 \right)^2 - ( 3 x )^2 \\ = \left\{ \left( x^2 + 5 \right) + 3 x \right\} \left\{ \left( x^2 + 5 \right) - 3 x \right\} = \left( x^2 + 3 x + 5 \right) \left( x^2 - 3 x + 5 \right)$

$x^4 - 11 x^2 + 49 = x^4 + 14 x^2 + 49 - 14 x^2 - 11 x^2 = \left( x^2 + 7 \right)^2 - 25 x^2 = \left( x^2 + 7 \right)^2 - \left( 5 x \right)^2 \\ = \left\{ \left( x^2 + 7 \right) + 5 x \right\} \left\{ \left( x^2 + 7 \right) - 5 x \right\} = \left( x^2 + 5 x + 7 \right) \left( x^2 - 5 x + 7 \right)$

$x^4 - 3 x^2 + 9 = x^4 + 6 x^2 + 9 - 6 x^2 - 3 x^2 = \left( x^2 + 3 \right)^2 - 9 x^2 = \left( x^2 + 3 \right)^2 - \left( 3 x \right)^2 \\ = \left\{ \left( x^2 + 3 \right) + 3 x \right\} \left\{ \left( x^2 + 3 \right) - 3 x \right\} = \left( x^2 + 3 x + 3 \right) \left( x^2 - 3 x + 3 \right)$

$x^4 - 15 x^2 + 9 = x^4 - 6 x^2 + 9 + 6 x^2 - 15 x^2 = \left( x^2 - 3 \right)^2 - 9 x^2 = \left( x^2 - 3 \right)^2 - \left( 3 x \right)^2 \\ = \left\{ \left( x^2 - 3 \right) + 3 x \right\} \left\{ \left( x^2 - 3 \right) - 3 x \right\} = \left( x^2 + 3 x - 3 \right) \left( x^2 - 3 x - 3 \right)$

$x^4 + 3 x^2 + 4 = x^4+ 4 x^2 + 4 - 4 x^2 + 3 x^2 = \left( x^2 + 2 \right)^2 - x^2 = \left\{ \left( x^2 + 2 \right) + x \right\} \left\{ \left( x^2 + 2 \right) - x \right\} \\ = \left( x^2 + x + 2 \right) \left( x^2 - x + 2 \right)$