Use the rational zeros theorem to find all the real zeros of the polynomial function. Use the zeros to factor f over the real numbers.

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

Possible rational zeros:

± 1, ± 1/2
f(1) = 2 - 1 + 2 - 1 ≠0
f(-1) = -2 - 1 - 2 - 1 ≠0
f(1/2) = 1/4 - 1/4 + 1-1 = 0 , yeahh, (2x - 1) is a factor
Using long division...

2x^3-x^2+2x-1 = (2x - 1)(x^2 + 1)

so x = 1/2 or x^2 = -1

x = 1/2 or x = ± i

so the only real zero is x=1/2

How do you use the real zeros to factor f?

if x=a is a real zero. (x-a) is a factor. Divide f(x) by (x-a) and see what the quotient is. Maybe you can factor it, maybe not. In this case, not.

Ok thank you

To find the real zeros of the polynomial function f(x) = 2x^3 - x^2 + 2x - 1 using the Rational Zeros Theorem, follow these steps:

Step 1: Write down all possible rational zeros.
The possible rational zeros are obtained by finding all possible factors of the constant term (in this case, -1) and dividing them by all possible 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, ±1/2

Step 2: Test the possible zeros using synthetic division.
Using synthetic division, divide the polynomial f(x) by each possible rational zero to test if it produces a remainder of 0.

Testing x = 1:
1 | 2 -1 2 -1
|<------2------|
|<------1------|
|<------3------|
|<------2------|
The remainder is 2, not 0.

Testing x = -1:
-1 | 2 -1 2 -1
|<--------2------|
|<-------3------|
|<--------1------|
|<---------0------|
The remainder is 0. Therefore, x = -1 is a zero.

Testing x = 1/2:
1/2 | 2 -1 2 -1
|<-------1------|
|<--------0------|
|<--------1/2-----|
|<---------0------|
The remainder is 0. Therefore, x = 1/2 is a zero.

Therefore, the real zeros of the polynomial f(x) = 2x^3 - x^2 + 2x - 1 are x = -1 and x = 1/2.

Step 3: Factor f(x) using the zeros.
Since x = -1 and x = 1/2 are zeros of the polynomial, we can factor f(x) using the factor theorem. The factors are obtained by subtracting the zeros from x as follows:

(x + 1)(x - 1/2)

Therefore, the factored form of f(x) over the real numbers is (x + 1)(x - 1/2).