use the rational zeros theorem to find all the real zeros of the polynomial. Use the zeros to factor f over the real numbers.
f(x)=2x^3-x^2+2x-1
One root is x = 1/2, so one factor is x - 1/2
Divide f(x) by x - 1/2 to yield a quadratic equation
2x^2 + x +2 = 0
that will provide the other two roots.
To use the Rational Zeros Theorem, we first need to determine the possible rational zeros of the polynomial f(x) = 2x^3 - x^2 + 2x - 1.
According to the Rational Zeros Theorem, the possible rational zeros are all the possible quotients of the factors of the constant term (-1) divided by the factors of the leading coefficient (2).
The factors of -1 are ±1, and the factors of 2 are ±1 and ±2.
Therefore, the possible rational zeros are:
±1/2, ±1, ±1/2.
Now, we will check if any of these possible zeros are actual zeros of the polynomial. We can do this by substituting each possible zero back into the polynomial and checking if the result is zero.
Substituting x = 1/2:
f(1/2) = 2(1/2)^3 - (1/2)^2 + 2(1/2) - 1 = 0
Substituting x = -1/2:
f(-1/2) = 2(-1/2)^3 - (-1/2)^2 + 2(-1/2) - 1 = 0
Thus, x = 1/2 and x = -1/2 are real zeros of the polynomial f(x) = 2x^3 - x^2 + 2x - 1.
To factor f(x) using these zeros, we can use long division or synthetic division to divide the polynomial by (x - 1/2)(x + 1/2). The resulting expression will be a quadratic, which can be factored easily.
Using synthetic division:
1/2 | 2 -1 2 -1
1 0 2 0
Therefore, the quadratic expression is obtained as 1x^2 + 2. This quadratic cannot be factored any further over the real numbers because it has no real roots.
Therefore, we can factor f(x) as:
f(x) = (x - 1/2)(x + 1/2)(x^2 + 2) over the real numbers.