Use the rationals theorem to find all the zeros of the polynomial function. Use the zeros to factor f over the real numbers.
f(x)=2x^3-x^2+2x-1
follow the same steps I just showed you in your last post
http://www.jiskha.com/display.cgi?id=1364332046
I like this variant which makes use of candidates that turn out not to be a zero. This works as follows.
The possible zeroes are:
x = -1,1,1/2,-1/2 (1)
We have:
f(1) = -2
If we put
g(t) = f(1+t)
then the coefficient of t^3 is 2 and the constant term is f(1) = -2, the possible zeroes are thus:
t = +/-1 , +/-2, +/- 1/2
The possible zeroes of f are thus:
x = 1+t = 0,2,-1,3,1/2,3/2
But since all the possible zeroes are listed in (1), we can strike out the elements that are not on that list. We are thus left with:
x = -1,1/2
Then since -1 is not a zero, the only possible rational zero is 1/2, which is indeed a zero. Then you can proceed by dividing f(x) by x-1/2 and find the zeroes of the quadratic.
To find the zeros of the polynomial function using the Rational Zero Theorem, we first need to identify all the possible rational zeros. This is done by creating a list of all the factors of the constant term (in this case, -1) and all the factors of the leading coefficient (in this case, 2).
The factors of -1 are ±1, and the factors of 2 are ±1 and ±2. Therefore, the possible rational zeros are ±1, ±1/2.
To check which of these possible zeros are actually zeros of the function, we can use synthetic division or long division. Let's start with x = 1:
1 | 2 -1 2 -1
| 2 1 3
____________
2 1 3 2
Since the remainder is not zero, x = 1 is not a zero of the function. Now let's try x = -1:
-1 | 2 -1 2 -1
| -2 3 -5
___________
2 -3 5 -6
Again, the remainder is not zero, so x = -1 is not a zero of the function. Next, let's try x = 1/2:
1/2 | 2 -1 2 -1
| 1/2 1 3/2
_____________
2 1/2 5/2 1/2
Once more, the remainder is not zero, so x = 1/2 is also not a zero of the function.
Therefore, there are no rational zeros for the given function f(x) = 2x^3 - x^2 + 2x - 1.
Since there are no rational zeros, the polynomial cannot be factored further over the real numbers. The polynomial f(x) = 2x^3 - x^2 + 2x - 1 is already in its factored form.