Algebra

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I am solving X^4-3x^2+2=0 I factored it to be (x^2-2)(x+1)(x-1) as my book suggested. But why is it three sets of parentheses? How do you know that it's supposed to be three sets of parentheses in opposed to two or four? And how come one of the answers according to my book is plus or minus the square root of two? I see why it might be plus the sq. rt. of two, but I don't get how it could be minus. Thank you for your help clearing this up!

  • Algebra -

    If you factorize the polynomial in two steps, you will see it more clearly.
    X^4-3x^2+2
    easily factorizes into two factors:
    (x²-2)(x²-1)
    Each of the above factors factorizes into two others by virtue of the identity
    (a²-b²)=(a+b)(a-b)
    The factor (x²-2) does not have a rational factor, but it can still be factored as
    (x²-2)=(x+√2)(x-√2)
    and the second one, as you have already done:
    (x²-1)=(x+1)(x-1)

    So the polynomial actually has four factors, two of which are irrational, and four corresponds to the degree of the polynomial.
    Since
    (x+√2)(x-√2)(x+1)(x-1)=0,
    it should be quite clear why the four zeroes are where they are.

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