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Math Help - how to factorize??

  1. #1
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    how to factorize??

    hi,
    how to factorize this expression?

    x^4 + x^2 + 1

    actually i know how to factorize, but would like to know there is only one answer to that. I first take out x^2 as common factor to factorize the three terms, then use the identities (a+b)^2 = a^2 + 2ab + b^2 and
    a^2 - b^2 = (a - b)(a + b) to factorize

    is there any other answer ? will the factorized expression be unique?



    2. Another question

    I really don't know how to factorize the following expression, after a long time of try

    I hope to understand the "techniques" of factorization, instead of just knowing how to do. Thanks all!!!!

    acx^2 + (ad + bc)xy - bdy^2

    the question is correctly typed, only x and y has power of 2












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  2. #2
    MHF Contributor red_dog's Avatar
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    Isn't it acx^2+(ad\textcolor{red}{-}bc)xy-bdy^2

    or acx^2+(ad+bc)xy\textcolor{red}{+}bdy^2 ?
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  3. #3
    ynj
    ynj is offline
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    Quote Originally Posted by louis1234 View Post
    hi,
    how to factorize this expression?

    x^4 + x^2 + 1

    actually i know how to factorize, but would like to know there is only one answer to that. I first take out x^2 as common factor to factorize the three terms, then use the identities (a+b)^2 = a^2 + 2ab + b^2 and
    a^2 - b^2 = (a - b)(a + b) to factorize

    is there any other answer ? will the factorized expression be unique?



    2. Another question

    I really don't know how to factorize the following expression, after a long time of try

    I hope to understand the "techniques" of factorization, instead of just knowing how to do. Thanks all!!!!

    acx^2 + (ad + bc)xy - bdy^2

    the question is correctly typed, only x and y has power of 2





    1.Actually, the "uniqueness"will always hold but the answer will be different. if you factorize it on rational number(parameter is rational), then the answer is yours. But in the real field, x^2+x+1=(x-\frac{-1-\sqrt{3}i}{2})(x-\frac{-1+\sqrt{3}i}{2}).
    P.S. In the factoralization of polynomial, two factors says to be "equivalent" if f(x)=dg(x)for some nonzero constant d,not exactly f(x)=g(x)!
    2.Same question as red_dog
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