因式分解 m^7-n^7x^7

解答

因式分解

m^{7} - n^{7} x^{7}

(m - n x) (m^{6} + m^{5} n x + m^{4} n^{2} x^{2} + m^{3} n^{3} x^{3} + m^{2} n^{4} x^{4} + m n^{5} x^{5} + n^{6} x^{6})

求解步骤

m^{7} - n^{7} x^{7}

将

n^{7} x^{7}

改写为

(n x)^{7}

= m^{7} - (n x)^{7}

使用因式分解法则

x^{n} - y^{n} = (x - y) (x^{n - 1} + x^{n - 2} y + \dots + x y^{n - 2} y^{n - 1})

m^{7} - (n x)^{7} = (m - n x) (m^{6} + n x m^{5} + (n x)^{2} m^{4} + (n x)^{3} m^{3} + (n x)^{4} m^{2} + (n x)^{5} m + (n x)^{6})

= (m - n x) (m^{6} + n x m^{5} + (n x)^{2} m^{4} + (n x)^{3} m^{3} + (n x)^{4} m^{2} + (n x)^{5} m + (n x)^{6})

化简

(m - n x) (m^{6} + n x m^{5} + (n x)^{2} m^{4} + (n x)^{3} m^{3} + (n x)^{4} m^{2} + (n x)^{5} m + (n x)^{6}) : (m - n x) (m^{6} + m^{5} n x + m^{4} n^{2} x^{2} + m^{3} n^{3} x^{3} + m^{2} n^{4} x^{4} + m n^{5} x^{5} + n^{6} x^{6})

= (m - n x) (m^{6} + m^{5} n x + m^{4} n^{2} x^{2} + m^{3} n^{3} x^{3} + m^{2} n^{4} x^{4} + m n^{5} x^{5} + n^{6} x^{6})

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