%2F(x-5).jpeg "(3x^2-14x-5)/(x-5)")
Simplifying Rational Expressions: (3x^2 - 14x - 5) / (x - 5)
This article will guide you through the process of simplifying the rational expression (3x^2 - 14x - 5) / (x - 5). We'll use techniques like factoring and cancellation to arrive at the simplest form.
Understanding Rational Expressions
A rational expression is a fraction where the numerator and denominator are both polynomials. Simplifying a rational expression means reducing it to its simplest form, where the numerator and denominator have no common factors other than 1.
Simplifying the Expression
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Factor the numerator:
- We need to find two numbers that multiply to -15 (3 * -5) and add up to -14. These numbers are -15 and 1.
- Therefore, we can factor the numerator as follows: (3x^2 - 14x - 5) = (3x + 1)(x - 5)
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Factor the denominator:
- The denominator is already in its simplest form.
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Cancel common factors:
- Notice that both the numerator and denominator share the factor (x - 5). We can cancel this factor: [(3x + 1)(x - 5)] / (x - 5) = 3x + 1
Conclusion
The simplified form of the rational expression (3x^2 - 14x - 5) / (x - 5) is 3x + 1. Remember that this simplified form is valid only when x ≠ 5, as the original expression is undefined at x = 5 due to the denominator being zero.