## 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.