(x-3)(x+5)(x-7)/ X-4 (x+6)

Jasonbradley

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Jun 17, 2024

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Analyzing the Rational Expression: (x-3)(x+5)(x-7) / (x-4)(x+6)

This expression represents a rational function, which is a function that can be expressed as the ratio of two polynomials. Understanding the properties of this expression can help us to analyze its behavior, identify its key features, and solve related problems.

Identifying Key Features:

1. Domain: The domain of a rational function is all real numbers except for the values that make the denominator zero. In this case:

• (x - 4) = 0 when x = 4
• (x + 6) = 0 when x = -6

Therefore, the domain of this function is all real numbers except for x = 4 and x = -6.

2. Vertical Asymptotes: Vertical asymptotes occur where the denominator becomes zero, but the numerator doesn't. This means:

• At x = 4 and x = -6, the function will have vertical asymptotes.

3. Horizontal Asymptote: The horizontal asymptote of a rational function depends on the degrees of the numerator and denominator. Here's how to determine it:

• Degree of numerator: 3 (since it's a product of three linear factors)
• Degree of denominator: 2 (since it's a product of two linear factors)

Since the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote. Instead, the function will exhibit oblique asymptotes.

4. x-intercepts (zeros): The x-intercepts occur where the numerator is zero. So:

• (x - 3) = 0 when x = 3
• (x + 5) = 0 when x = -5
• (x - 7) = 0 when x = 7

The function has x-intercepts at x = 3, x = -5, and x = 7.

5. y-intercept: The y-intercept occurs when x = 0. Plugging in x = 0:

• (0 - 3)(0 + 5)(0 - 7) / (0 - 4)(0 + 6) = 105 / 24 = 35/8

Therefore, the function has a y-intercept at (0, 35/8).

Further Analysis:

• Oblique Asymptotes: To find the oblique asymptote, we can perform long division or synthetic division. The result will be a linear function that represents the oblique asymptote.

• End Behavior: As x approaches positive or negative infinity, the function will approach the oblique asymptote.

• Holes: There are no common factors between the numerator and denominator, so there are no holes in the graph.

Conclusion:

This analysis gives us a comprehensive understanding of the behavior of the rational function represented by the expression (x-3)(x+5)(x-7) / (x-4)(x+6). By knowing its domain, asymptotes, intercepts, and end behavior, we can accurately sketch its graph and understand its properties.