%2F(x-3).jpeg "(2x^2-5x-3)/(x-3)")
Simplifying Rational Expressions: (2x^2 - 5x - 3) / (x - 3)
This article will explore how to simplify the rational expression (2x^2 - 5x - 3) / (x - 3).
Understanding Rational Expressions
Rational expressions are fractions where the numerator and denominator are polynomials. Simplifying rational expressions often involves factoring the numerator and denominator to identify common factors that can be canceled.
Factoring the Numerator
First, we need to factor the quadratic expression in the numerator (2x^2 - 5x - 3). We can achieve this by finding two numbers that multiply to give -6 (2 times -3) and add up to -5. These numbers are -6 and 1.
- (2x^2 - 5x - 3) = (2x + 1)(x - 3)
Cancellation of Common Factors
Now we have: (2x + 1)(x - 3) / (x - 3)
We can cancel out the common factor (x - 3) from the numerator and denominator.
Simplified Expression
This leaves us with the simplified expression: 2x + 1.
Restrictions
It's important to note that the original expression is undefined when x = 3. This is because it would lead to division by zero. Therefore, the simplified expression 2x + 1 is only valid for x ≠ 3.
Conclusion
Simplifying the rational expression (2x^2 - 5x - 3) / (x - 3) involves factoring the numerator, canceling common factors, and understanding the restrictions. The final simplified expression is 2x + 1, valid for x ≠ 3.