How do you use the rational roots theorem to find all possible zeros of #f(x) = 6x^4 + 4x^3 - 2x^2 + 2#?

1 Answer
Mar 31, 2016

Answer:

There are no Real zeros for the given equation.

Explanation:

If we attempt to use the Rational Root Theorem by testing
#color(white)("XXX")x=+-abs("factors of " 2)/abs("factors of " 6)#
in #f(x)=6x^44x^3-2x^2+2#
the candidate values would be:
#color(white)("XXX")+-{1,1/2,1/3,1/6,cancel(1/2),cancel(2/2),2/3,cancel(2/6)}#
but as indicated in the table below, none of these give a zero result:

enter image source here
Therefore there are no rational zeros for this expression.

In fact, if we consider the graph for this expression, we can see that it has no Real zeros.
graph{6x^4+4x^3-2x+2 [-8.036, 9.754, -0.84, 8.05]}

#6x^4+4x^3-2x^2+2# should have #4# imaginary zeroes, but they can not be determined using the Rational Factor Theorem.