How do you find all possible rational zeros of #f(x) = 2x^3 - 5x^2 + 3x - 1#?

1 Answer
May 27, 2016

Answer:

Use the rational root theorem to help find that it has no rational zeros.

Explanation:

#f(x) = 2x^3-5x^2+3x-1#

By the rational root theorem, any rational zeros of #f(x)# are expressible in the form #p/q# for integers #p, q# with #p# a divisor of the constant term #-1# and #q# a divisor of the coefficient #2# of the leading term.

That means that the only possible rational zeros are:

#+-1/2#, #+-1#

In addition, note that there are no changes of signs of coefficients in #f(-x) = -2x^3-5x^2-3x-1#, so #f(x)# has no negative zeros.

That leaves possible rational zeros:

#1/2#, #1#

Then we find:

#f(1/2) = 1/4-5/4+3/2-1 = -1/2#

#f(1) = 2-5+3-1 = -1#

So this cubic has no rational zeros.