## Simplifying Rational Expressions: (8x^3-3x+1)/(4x^3+x^2-2x-3)

This article explores the simplification of the rational expression (8x^3-3x+1)/(4x^3+x^2-2x-3).

### Understanding Rational Expressions

A rational expression is a fraction where the numerator and denominator are polynomials. To simplify a rational expression, we aim to reduce it to its lowest terms by factoring both the numerator and denominator and canceling out any common factors.

### Factoring the Expression

**1. Factoring the Numerator:**

The numerator (8x^3-3x+1) can be factored using the Rational Root Theorem and synthetic division.

**Rational Root Theorem:**Potential rational roots are factors of the constant term (1) divided by factors of the leading coefficient (8). This gives us ±1, ±1/2, ±1/4, ±1/8.**Synthetic Division:**Testing these potential roots, we find that x=1/2 is a root.

Performing synthetic division with x=1/2, we get:

```
2x^2 + x - 2
x=1/2 | 8 0 -3 1
4 2 -1
-----------------
8 4 -1 0
```

Therefore, the factored form of the numerator is (2x^2 + x - 2)(2x - 1).

**2. Factoring the Denominator:**

The denominator (4x^3 + x^2 - 2x - 3) can be factored using grouping.

**Grouping:**Group the first two terms and the last two terms: (4x^3 + x^2) + (-2x - 3)**Factor out common factors:**x^2(4x + 1) - 1(2x + 3)**Factor by grouping:**(x^2 - 1)(4x + 1)**Difference of Squares:**(x+1)(x-1)(4x+1)

### Simplifying the Expression

Now, we can rewrite the original expression with its factored form:

[(2x^2 + x - 2)(2x - 1)] / [(x+1)(x-1)(4x+1)]

Notice that there are no common factors between the numerator and denominator. Therefore, the expression cannot be simplified further.

### Conclusion

The simplified form of the rational expression (8x^3-3x+1)/(4x^3+x^2-2x-3) is:

**[(2x^2 + x - 2)(2x - 1)] / [(x+1)(x-1)(4x+1)]**

This expression cannot be simplified further, as there are no common factors between the numerator and denominator.