Algebra 2

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Hey how would you factor this polynomial to find the zeroes of this function?

x^4-5x^3+7x^2+3x-10

  • Algebra 2 -

    Always try x = 1 and x = -1
    If x = 1 the function is -4
    but if x = -1 the function is zero.
    Therefore (x+1) is a factor, so use long division.
    (x+1)(x^3-6x^2+13x-10)
    Now we need something that works with 10 like 10 and 1 or 5 and 2
    Try 2 and -2
    2 works so (x-2) is a factor. Divide
    (x^3-6x^2+13x-10) by (x-2)
    (x+1)(x-2)(x^2-4x+5)
    the last one you must do by quadratic equation and the roots are complex
    x = (2+i) and x = (2-i)
    so
    (x+1)(x-2)(x-2+i)(x-2-i)
    and
    x = -1, 2 , 2+i, 2-i

  • Algebra 2 -

    Try +/- 1,2,5 and 10, the integer factors of 10. That will apply the "rational roots" theorem.
    You will see right away that x = -1 makes the polynomial zero. Therefore x+1 is a factor.

    Divide x^4-5x^3+7x^2+3x-10 by x+1 for the other factor, a cubic polynomial.
    That cubic factor is
    x^3 -6x^2 +13x -10
    x=2 makes that zero, so x-2 is a factor.
    Divide x^3 -6x^2 +13x -10 by x-2 to get a quadratic factor. It will be
    x^2 -4x + 5
    Then see if you can factor that. The remaining roots are complex.

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