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11 July, 17:14

Factor completely x^8-1/81

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  1. 11 July, 17:28
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    Remember the difference of 2 perfect squares

    (a²-b²) = (a-b) (a+b)

    1/81 = (1/9) ²

    so

    x⁸ = (x⁴) ²

    x⁸-1/81 = (x⁴) ² - (1/9) ² = (x⁴-1/9) (x⁴+1/9)

    oh look, 1/9 = (1/3) ²

    x⁴ = (x²) ²

    we can factor that first one again

    x⁴-1/9 = (x²) ² - (1/3) ² = (x²-1/3) (x²+1/3)

    so factored completely is (x²-1/3) (x²+1/3) (x⁴+1/9)

    you could go further and force taht first term to be another difference by doing 1/3 = (√ (1/3)) ², but I wouldn't

    (x²-1/3) (x²+1/3) (x⁴+1/9) is factored
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