%2F(x-2).jpeg "(2x^2+x-10)/(x-2)")
Analyzing the Expression (2x^2 + x - 10) / (x - 2)
This expression represents a rational function, a function that is defined as a ratio of two polynomials.
Understanding the Components
- Numerator: 2x² + x - 10
- Denominator: x - 2
Simplifying the Expression
We can simplify this expression by performing polynomial division. This process involves dividing the numerator by the denominator.
Step 1: Set up the long division.
2x + 5
x - 2 | 2x² + x - 10
-(2x² - 4x)
5x - 10
-(5x - 10)
0
Step 2: The result of the division is 2x + 5. Therefore, we can rewrite the expression as:
(2x² + x - 10) / (x - 2) = 2x + 5
Important Note: This simplification is valid only for values of x where x ≠ 2. This is because the original expression is undefined when x = 2, as it results in division by zero.
Analyzing the Simplified Expression
The simplified expression, 2x + 5, represents a linear function. This means it has a constant rate of change and a straight line graph.
Key Characteristics:
- Slope: 2
- Y-intercept: 5
Implications of the Simplification
The simplification process reveals that the original rational function, (2x² + x - 10) / (x - 2), behaves identically to the linear function 2x + 5, except for the point x = 2. At x = 2, the original function has a vertical asymptote, which means the function approaches infinity as x approaches 2.
Visual Representation
A graph of the original function will show a curve that closely resembles the line 2x + 5, except for a gap at x = 2 where the vertical asymptote exists.
Conclusion
Understanding the components and simplifying the expression (2x² + x - 10) / (x - 2) allows us to gain insights into its behavior and its relationship to the linear function 2x + 5. This knowledge is crucial for analyzing, interpreting, and applying this expression in various mathematical and real-world contexts.