%2F(x-5).jpeg "(2x^2+x-7)/(x-5)")
Simplifying the Rational Expression (2x^2 + x - 7)/(x - 5)
This article will guide you through the process of simplifying the rational expression (2x^2 + x - 7)/(x - 5).
Understanding Rational Expressions
A rational expression is a fraction where the numerator and denominator are both polynomials. To simplify a rational expression, we aim to factor both the numerator and denominator and then cancel out any common factors.
Steps to Simplify (2x^2 + x - 7)/(x - 5)
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Factor the numerator: We need to find two numbers that multiply to -14 (2 * -7) and add up to 1 (the coefficient of the x term). The numbers 7 and -2 satisfy these conditions. Therefore:
2x² + x - 7 = (2x - 7)(x + 1)
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Factor the denominator: The denominator is already in its simplest factored form: (x - 5).
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Simplify by canceling common factors: Our expression now looks like this: [(2x - 7)(x + 1)] / (x - 5). There are no common factors to cancel out between the numerator and denominator.
The Simplified Expression
Therefore, the simplified form of the rational expression (2x² + x - 7)/(x - 5) is (2x - 7)(x + 1) / (x - 5).
Important Note
Remember that this simplified expression is equivalent to the original expression only if x ≠ 5. This is because the original expression is undefined when x = 5, as it leads to division by zero.