How can you use the rational zeros theorem with this problem to find all real zeros??
#f(x)=2x^3-x^2+2x-1#
2 Answers
The only real zero is
Explanation:
Given:
#f(x) = 2x^3-x^2+2x-1#
By the rational roots theorem, any rational zeros of
So the only possible rational zeros are:
#+-1/2# ,#+-1#
In addition, note that the pattern of the signs of the coefficients is
So the only possible rational zeros are:
#1/2# ,#1#
We find:
#f(1/2) = 2(1/8)-(1/4)+2(1/2)-1 = 0#
#f(1) = 2-1+2-1 = 2#
So the only rational real zero is
This has a corresponding factor
#2x^3-x^2+2x-1 = (2x-1)(x^2+1)#
Note that
There is only one Real zero occurs at:
Explanation:
According to the Rational Zero Theorem:
For a polynomial of the standard form
any rational zero must be of the form
where
and
For the given polynomial:
this means that the only candidates as possible rational zeros are
Evaluating the given function for each of these values:
So the only rational zero is at
Note that the Rational Zero Theorem does not indicate if there are or are not non-rational, Real zeros.
To determine this for the given case:
Note that if
Performing the division:
So (for Real values) if
then
But
which has no Real solutions.
Therefore (among Real solutions)