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I need to find the critical number of 4x^3-36x^2+96x-64 by factoring. Basically I need to know the zeros.

  • calculus -

    If you plug in x=1, you find that f(x)=4x^3-36x^2+96x-64 evaluates to zero. From here, use either synthetic division or long division to get the quadratic 4x^2-32x+64. Reduce to x^2-8x+16 and solve by factoring into (x-4)^2.
    So 4x^3-36x^2+96x-64=4(x-1)(x-4)^2.

  • calculus -

    The critical numbers are the values for x in which the function has a horizontal tangent line, so where the first derivative is zero.
    0= x^2-6x+8
    The critical numbers are
    x=4 and x=2

  • calculus -

    If you need to find the zeros of a cubic by factoring then the resault is:

    4(x-4)^2-(x-1), so the zeros are at 4, and 1, and these are also the critical points.

    You did not specify if 4x^3-36x^2+96x-64 was your function, or the first derivative of the function, however if it is not then you should take the first derivative which should leave you a quadratic function, then use the quadratic formula on the remaining second-degree polynomial.

  • calculus -

    Well, everything divides by 4, so you can take that out for a start, leaving

    x^3 - 9x^2 + 24x - 16

    We need three numbers with a product of -16 (1, 2, 4, 8 and 16 are the only possibilities) that sum to -9.

    Shouldn't be too hard to figure out from there.

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