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受欢迎的代数 >

因式分解 (a^2-1)^3-27

解答

因式分解

(a^{2} - 1)^{3} - 27

解答

(a + 2) (a - 2) (a^{4} + a^{2} + 7)

求解步骤

(a^{2} - 1)^{3} - 27

将

27

改写为

3^{3}

= (a^{2} - 1)^{3} - 3^{3}

使用立方差公式:

x^{3} - y^{3} = (x - y) (x^{2} + x y + y^{2})

(a^{2} - 1)^{3} - 3^{3} = ((a^{2} - 1) - 3) ((a^{2} - 1)^{2} + 3 (a^{2} - 1) + 3^{2})

= ((a^{2} - 1) - 3) ((a^{2} - 1)^{2} + 3 (a^{2} - 1) + 3^{2})

整理后得

= (a^{2} - 4) ((a^{2} - 1)^{2} + 3 (a^{2} - 1) + 9)

分解

a^{2} - 4 : (a + 2) (a - 2)

= (a + 2) (a - 2) ((a^{2} - 1)^{2} + 3 (a^{2} - 1) + 9)

(a^{2} - 1)^{2} = (a + 1)^{2} (a - 1)^{2}

3 (a^{2} - 1) = 3 (a + 1) (a - 1)

= (a + 2) (a - 2) ((a + 1)^{2} (a - 1)^{2} + 3 (a + 1) (a - 1) + 9)

乘开

(a + 1)^{2} (a - 1)^{2} + 3 (a + 1) (a - 1) + 9 : a^{4} + a^{2} + 7

= (a + 2) (a - 2) (a^{4} + a^{2} + 7)

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