%2F(2x%5E3-4x%5E2-x%2B2).jpeg "(10x^3+5x^2-1)/(2x^3-4x^2-x+2)")
Simplifying Rational Expressions: A Step-by-Step Guide
This article will guide you through the process of simplifying the rational expression:
(10x^3 + 5x^2 - 1) / (2x^3 - 4x^2 - x + 2)
1. Factoring the Numerator and Denominator
The first step is to factor both the numerator and the denominator. This will help us identify any common factors that can be cancelled out.
Numerator:
- Factor out a common factor of 5: 5(2x^3 + x^2 - 1/5)
Denominator:
- Factor by grouping:
- (2x^3 - 4x^2) + (-x + 2)
- 2x^2(x - 2) - 1(x - 2)
- (2x^2 - 1)(x - 2)
2. Identifying and Cancelling Common Factors
Now that both the numerator and denominator are factored, we can look for common factors. In this case, we can see that there are no common factors between the two expressions.
3. Simplifying the Expression
Since there are no common factors, the expression cannot be simplified further.
4. Final Result
The simplified form of the rational expression is:
(10x^3 + 5x^2 - 1) / (2x^3 - 4x^2 - x + 2) = 5(2x^3 + x^2 - 1/5) / [(2x^2 - 1)(x - 2)]
Note: This expression cannot be simplified further, even though the numerator and denominator are factored.