Rational Zero Theorem?

Question

Can someone explain how to use the Rational Zeros Theorem to find all of the real zeros of the the following polynomials functions?

Use the zeros to factor f over the real numbers

f(x) = x^4= -7x^2 -144

and

f(x) = x^4 -24x^2- 25 I believe the answer to this one is is  -5,5; f(x) = (x-5) (x+5)(x^2+1)

I just don't understand how to use the theorem to solve this. Can anyone help? Thanks

Answer 1

both of your equations can be factored

x^4-7x^2-144 =(x^2-16)(x^2 +9) =(x-4)(x+4)(x^2+9)

giving you 4 and -4 as real roots

the second one factors into (x^2-25)(x^2+1) =(x-5)(x+5)(x^2+1)

giving you 5 and -5

when you have only x^4, x^2 and a constant, you can factor as above just like

ax^2 +bx+c

The rational root theorem is a much longer process

you would have to try all the positive and negative factors (using synthetic division) of 144 to try and find rational roots