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rational zeros theorem

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Use the rational zeros theorem to find all the real zeros of the polynomial function. Use the zeros to factor f over the real numbers.
Find the real zeros
Use the real zeros to factor f

  • rational zeros theorem -

    only possible rational roots are
    x = ±1, ±1/5, ± 1/25

    quickly found x=-1 to work
    so one factor is x+1

    after reducing it to a cubic by synthetic division, it took a bit longer to find x = -1/25 to work
    so (25x+1) is another factor
    long algebraic divsion gave the last factor as x^2 + 5, which has no real roots.

    so real roots are
    x = -1 and x = -1/25

  • rational zeros theorem -

    just a note:
    Things worked out in this case, but x = ±5 were also candidates, since 5/1 has suitable numerator and denominator. For example,

    has similar coefficients, but has real roots -1 and 5:

  • rational zeros theorem -

    given that f(x) = 9/x-5 and g(x) = 12/x+12 find

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