%5E2(x%2B1)(x-8)%5E2.jpeg "(x+6)^2(x+1)(x-8)^2")
Analyzing the Polynomial: (x+6)^2(x+1)(x-8)^2
This polynomial is a product of four factors:
- (x+6)^2: This factor contributes a double root at x = -6.
- (x+1): This factor contributes a single root at x = -1.
- (x-8)^2: This factor contributes a double root at x = 8.
Let's explore the characteristics of this polynomial:
Roots and Multiplicity
The roots of a polynomial are the values of x that make the polynomial equal to zero. The multiplicity of a root tells us how many times the corresponding factor appears in the polynomial.
Here's a table summarizing the roots and their multiplicities:
| Root | Multiplicity |
|---|---|
| -6 | 2 |
| -1 | 1 |
| 8 | 2 |
Key Point: The multiplicity of a root affects the behavior of the graph at that point.
- Odd Multiplicity: The graph crosses the x-axis at the root.
- Even Multiplicity: The graph touches the x-axis at the root but does not cross it.
End Behavior
The end behavior of a polynomial is determined by its leading term. To find the leading term, we multiply the leading terms of each factor:
(x+6)^2(x+1)(x-8)^2 = (x^2)(x)(x^2) = x^5
Since the leading term has an odd degree and a positive coefficient, the end behavior is:
- As x approaches negative infinity, y approaches negative infinity.
- As x approaches positive infinity, y approaches positive infinity.
Graphing the Polynomial
Based on the information above, we can sketch a rough graph of the polynomial.
- Mark the roots: Plot the points (-6, 0), (-1, 0), and (8, 0) on the x-axis.
- Consider multiplicity: The graph will touch but not cross the x-axis at x = -6 and x = 8 (due to even multiplicity). The graph will cross the x-axis at x = -1 (due to odd multiplicity).
- End behavior: The graph will start in the bottom left quadrant and end in the top right quadrant.
Important Note: This is a rough sketch. The exact shape of the graph would require a more detailed analysis.
Further Exploration
To fully understand the behavior of this polynomial, we could also analyze:
- Turning points: The number of turning points in a polynomial is at most one less than its degree.
- Intervals of increasing and decreasing: We can identify the intervals where the function is increasing or decreasing.
- Local maxima and minima: These are the points where the function reaches a peak or valley.
This polynomial provides a good example of how the factors and their multiplicities contribute to the behavior of a function. Understanding these concepts is essential for analyzing and graphing polynomials.