# What are all the possible rational zeros for #f(x)=2x^3-3x^2-4x+6# and how do you find all zeros?

##### 1 Answer

Possible Rational Zeros:

Actual Zeros:

#### Explanation:

The way to determine all possible roots is through The Rational Roots Theorem, which states:

If P(x) is a polynomial with integer coefficients and if

Basically, find all integer (positive AND negative) factors of both the constant term (6, in this case) and the leading coefficient (2, in this case), and find all possible quotients

The integer factors of 6 are:

The integer factors of 2 are:

Therefore, the possible rational roots are:

To find which of these are roots in the actual equation, you could use Guess and Check with Synthetic Division, or simply group and factor the polynomial. I'll show the latter for the sake of space preservation:

From this we can conclude that one rational root is

However, we are still stuck with

So your complete factorization of this polynomial is:

#f(x) = (2x - 3)(x - sqrt2)(x + sqrt2)

From this, we can determine that the roots of this equation are