%2F(x%5E3-27).jpeg "(x^2-7x+12)/(x^3-27)")
Simplifying and Analyzing the Rational Expression: (x^2 - 7x + 12) / (x^3 - 27)
This article explores the rational expression (x^2 - 7x + 12) / (x^3 - 27), covering its simplification, domain, and key characteristics.
Simplifying the Expression
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Factor the numerator and denominator:
- The numerator factors as (x - 3)(x - 4)
- The denominator is a difference of cubes, which factors as (x - 3)(x^2 + 3x + 9)
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Cancel common factors:
- Both numerator and denominator share a factor of (x - 3).
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Simplified expression:
- The simplified expression is (x - 4) / (x^2 + 3x + 9)
Domain of the Expression
The domain of a rational expression is all real numbers except for values that make the denominator zero.
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Setting the denominator to zero: x^2 + 3x + 9 = 0
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Solving the quadratic equation: This equation has no real roots. Therefore, the denominator is never zero.
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Conclusion: The domain of the expression is all real numbers, (-∞, ∞).
Key Characteristics
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Vertical Asymptotes:
- Since the denominator has no real roots, the expression does not have any vertical asymptotes.
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Horizontal Asymptotes:
- The degree of the numerator (1) is less than the degree of the denominator (2). This indicates that the expression has a horizontal asymptote at y = 0.
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Holes:
- We canceled the common factor (x - 3). This indicates a hole in the graph at x = 3. To find the y-coordinate of the hole, substitute x = 3 into the simplified expression: (3 - 4) / (3^2 + 3(3) + 9) = -1/27. Therefore, the hole is located at (3, -1/27).
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x-intercept:
- The x-intercept occurs when the numerator equals zero. This happens when x = 4. Therefore, the x-intercept is (4, 0).
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y-intercept:
- The y-intercept occurs when x = 0. Substitute x = 0 into the simplified expression: (0 - 4) / (0^2 + 3(0) + 9) = -4/9. Therefore, the y-intercept is (0, -4/9).
Conclusion
The simplified rational expression (x - 4) / (x^2 + 3x + 9) has a domain of all real numbers, a horizontal asymptote at y = 0, a hole at (3, -1/27), an x-intercept at (4, 0), and a y-intercept at (0, -4/9). Understanding these characteristics helps to accurately visualize and analyze the behavior of the expression.