rational zeros theorem

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.
f(x)=25x^4+26x^3+126x^2+130x+5
Find the real zeros
x=
Use the real zeros to factor f
f(x)=

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  1. 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

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  2. 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,

    25x^4-100x^3-124x^2-4x-5
    has similar coefficients, but has real roots -1 and 5:
    (x+1)(x-5)(25x^2+1)

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

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