%2F(x-3).jpeg "(x^2+6x+12)/(x-3)")
Analyzing the Rational Expression (x^2 + 6x + 12) / (x - 3)
This article explores the rational expression (x^2 + 6x + 12) / (x - 3), covering its key features, simplification, and potential applications.
Understanding the Expression
The expression (x^2 + 6x + 12) / (x - 3) represents a rational function, which is a function defined as a ratio of two polynomials.
- Numerator: x^2 + 6x + 12 is a quadratic polynomial.
- Denominator: x - 3 is a linear polynomial.
Simplifying the Expression
The expression cannot be simplified further by factoring. This means the numerator and denominator do not share any common factors.
Analyzing the Expression
1. Domain: The function is defined for all real numbers except where the denominator equals zero.
- x - 3 = 0
- x = 3
Therefore, the domain of the function is all real numbers except x = 3. This value is referred to as a vertical asymptote.
2. Vertical Asymptote: As x approaches 3, the denominator approaches zero, making the function approach infinity. This indicates a vertical asymptote at x = 3.
3. Horizontal Asymptote: To find the horizontal asymptote, we compare the degrees of the numerator and denominator:
- The degree of the numerator (2) is greater than the degree of the denominator (1).
This indicates that there is no horizontal asymptote. The function will approach infinity as x approaches positive or negative infinity.
4. Intercepts:
- x-intercept: To find the x-intercept, set the numerator equal to zero and solve:
- x^2 + 6x + 12 = 0
- This quadratic equation does not have real roots. Therefore, there are no x-intercepts.
- y-intercept: To find the y-intercept, set x = 0:
- (0^2 + 6(0) + 12) / (0 - 3) = -4
Therefore, the y-intercept is (0, -4).
Graphing the Function
The graph of the function will exhibit the following characteristics:
- Vertical asymptote at x = 3
- No horizontal asymptote
- y-intercept at (0, -4)
- No x-intercepts
The graph will approach infinity as x approaches 3 and infinity as x approaches positive or negative infinity.
Applications
Rational functions like (x^2 + 6x + 12) / (x - 3) can be used to model various real-world phenomena:
- Physics: Describing the motion of objects under certain conditions.
- Economics: Modeling supply and demand curves.
- Engineering: Analyzing the behavior of circuits and systems.
Understanding the characteristics and behavior of this type of function is crucial for applying them to solve practical problems in various fields.