What are all the possible rational zeros for #f(x)=4x^4+19x^2-63# and how do you find all zeros?
1 Answer
Explanation:
By the rational roots theorem, any rational zeros of
That means that the only possible rational zeros are:
#+-1/4, +-1/2, +-3/4, +-1, +-3/2, +-7/4, +-9/4, +-3, +-7/2, +-9/2, +-21/4, +-7, +-9, +-21/2, +-63/4, +-21, +-63/2, +-63#
We could try each of these in turn, but there are easier ways to find the zeros of
Note that
#x^2 = (-19+-sqrt(19^2-4(4)(-63)))/(2*4)#
#=(-19+-sqrt(361+1008))/8#
#=(-19+-sqrt(1369))/8#
#=(-19+-37)/8#
i.e.
Since the result is rational, we could have found this using an AC method instead, but at least the quadratic formula gives it to us directly.
Hence