%2F(x-3).jpeg "(2x^2+5x-3)/(x-3)")
Simplifying the Rational Expression: (2x^2 + 5x - 3) / (x - 3)
This article will guide you through simplifying the rational expression (2x² + 5x - 3) / (x - 3).
Understanding Rational Expressions
A rational expression is a fraction where the numerator and denominator are polynomials. To simplify these expressions, we look for common factors that can be canceled out.
Factoring the Numerator
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Find two numbers that multiply to -6 (2 x -3) and add up to 5. These numbers are 6 and -1.
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Rewrite the middle term (5x) using these two numbers.
(2x² + 6x - x - 3) / (x - 3) -
Factor by grouping:
- Group the first two terms and the last two terms: (2x² + 6x) + (-x - 3)
- Factor out the greatest common factor from each group: 2x(x + 3) - 1(x + 3)
- Factor out the common binomial: (x + 3)(2x - 1)
Simplifying the Expression
Now we have: [(x + 3)(2x - 1)] / (x - 3)
Since there are no common factors in the numerator and denominator, this is the simplified form of the expression.
Important Note
It is crucial to remember that this expression is undefined when x = 3. This is because the denominator would become zero, resulting in an undefined value.
Conclusion
The simplified form of (2x² + 5x - 3) / (x - 3) is (x + 3)(2x - 1) / (x - 3), where x ≠ 3. This simplification process involves factoring the numerator and looking for common factors to cancel out.