## Factoring and Solving the Polynomial: (x+4)(x+2)(x-1)(x-3)(x-5)

This article explores the factored polynomial (x+4)(x+2)(x-1)(x-3)(x-5) and delves into its properties, solutions, and potential applications.

### Understanding the Factored Form

The given polynomial is already in factored form, making it easier to identify its roots and analyze its behavior. Each factor represents a linear expression that equals zero when x takes on a specific value.

**(x + 4):**This factor becomes zero when x = -4.**(x + 2):**This factor becomes zero when x = -2.**(x - 1):**This factor becomes zero when x = 1.**(x - 3):**This factor becomes zero when x = 3.**(x - 5):**This factor becomes zero when x = 5.

### Finding the Roots (Solutions)

The roots of a polynomial are the values of x that make the polynomial equal to zero. Since the polynomial is already factored, finding the roots is straightforward. We simply set each factor to zero and solve for x. Therefore, the roots of this polynomial are:

**x = -4****x = -2****x = 1****x = 3****x = 5**

These roots represent the x-intercepts of the graph of the polynomial.

### Expanding the Polynomial

To gain a deeper understanding of the polynomial's behavior, we can expand the factored form:

(x + 4)(x + 2)(x - 1)(x - 3)(x - 5) = (x² + 6x + 8)(x² - 4x + 3)(x - 5)

Further expansion would yield a fifth-degree polynomial.

### Applications

Polynomials like this can be used in various fields such as:

**Physics:**Modeling trajectories, oscillations, and wave phenomena.**Engineering:**Designing structures, analyzing circuits, and optimizing processes.**Economics:**Studying economic models and predicting market behavior.

### Key Takeaways

- The polynomial (x+4)(x+2)(x-1)(x-3)(x-5) is a fifth-degree polynomial.
- Its roots are -4, -2, 1, 3, and 5.
- The factored form provides a clear understanding of the polynomial's roots.
- Expanding the polynomial reveals its more complex form.

This analysis provides a fundamental understanding of the polynomial (x+4)(x+2)(x-1)(x-3)(x-5), its roots, and its potential applications in various fields. Further exploration of its graph, derivatives, and applications in specific contexts can provide even deeper insights.