rational zeros theorem
posted by Jill on .
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)=

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 
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^4100x^3124x^24x5
has similar coefficients, but has real roots 1 and 5:
(x+1)(x5)(25x^2+1) 
given that f(x) = 9/x5 and g(x) = 12/x+12 find