(8x^3-3x+1)/(4x^3+x^2-2x-3)

3 min read Jun 16, 2024

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.