%2F(2x%2B1).jpeg "(6x^2-7x-5)/(2x+1)")
Simplifying the Rational Expression: (6x² - 7x - 5) / (2x + 1)
This article will guide you through the process of simplifying the rational expression (6x² - 7x - 5) / (2x + 1).
Understanding Rational Expressions
A rational expression is simply a fraction where the numerator and denominator are polynomials. To simplify a rational expression, we aim to factor both the numerator and denominator and cancel out any common factors.
Factoring the Numerator
- Find two numbers that multiply to give -30 (the product of the leading coefficient and the constant term) and add up to -7 (the coefficient of the middle term). These numbers are -10 and 3.
- Rewrite the middle term (-7x) as the sum of these two numbers. This gives us: 6x² - 10x + 3x - 5.
- Factor by grouping:
- Group the first two terms and the last two terms: (6x² - 10x) + (3x - 5).
- Factor out the greatest common factor (GCF) from each group: 2x(3x - 5) + 1(3x - 5).
- Notice that both terms now have a common factor of (3x - 5). Factor this out: (3x - 5)(2x + 1).
Simplifying the Expression
Now we have: (6x² - 7x - 5) / (2x + 1) = (3x - 5)(2x + 1) / (2x + 1)
Since (2x + 1) appears in both the numerator and denominator, we can cancel them out.
This leaves us with the simplified expression: 3x - 5
Conclusion
Therefore, the simplified form of the rational expression (6x² - 7x - 5) / (2x + 1) is 3x - 5, provided that x ≠ -1/2 (as this would make the denominator zero). Remember, simplifying rational expressions often involves factoring and canceling out common factors to reach the most concise form.