Factorizing x^3 + 3x - 2

To factorize the given expression x^3 + 3x - 2, we need to find the factors that can be multiplied together to obtain the original expression. In this case, we are looking for a product of two binomials.

Step 1: Try to find a rational root using the Rational Root Theorem

The Rational Root Theorem states that if a polynomial has a rational root, then it can be expressed as the quotient of two factors: p/q, where p is a factor of the constant term (in this case, -2) and q is a factor of the leading coefficient (in this case, 1).

In this case, the factors of -2 are ±1 and ±2, and the factors of 1 are ±1. Therefore, the possible rational roots are ±1 and ±2.

Step 2: Test the possible rational roots using synthetic division

Using synthetic division, we can test each possible rational root to see if it is a factor of the given expression.

Let's start with x = 1:
1 | 1 0 3 -2
| 1 1 4
------------
1 1 4 2

Since the remainder is not zero, x = 1 is not a factor.

Next, let's try x = -1:
-1 | 1 0 3 -2
| -1 1 -4
-------------
1 -1 4 -6

Again, the remainder is not zero, so x = -1 is not a factor.

We can continue this process for x = 2 and x = -2, but we can save time by noticing that none of the possible rational roots are factors of the given expression.

Step 3: Apply the Factor Theorem

Since none of the rational roots are factors, we can conclude that the given expression is irreducible over the rational numbers. In other words, it cannot be factored into linear factors with rational coefficients.

Therefore, the factorization of x^3 + 3x - 2 is x^3 + 3x - 2 (irreducible over the rational numbers).