(x-1)(x+5)(x+3)

4 min read Jun 17, 2024

Exploring the Cubic Polynomial: (x-1)(x+5)(x+3)

This article delves into the fascinating world of the cubic polynomial represented by the expression (x-1)(x+5)(x+3). We will examine its key features, explore its graphical representation, and discuss its real-world applications.

Understanding the Factorized Form

The given expression is already in its factored form, which is incredibly useful for understanding the polynomial's behavior. Each factor corresponds to a root (or x-intercept) of the function.

(x-1): This factor indicates that the polynomial has a root at x = 1.
(x+5): This factor indicates a root at x = -5.
(x+3): This factor indicates a root at x = -3.

Expanding the Expression

While the factored form is insightful, expanding the expression provides a different perspective. Let's multiply the factors:

(x-1)(x+5)(x+3) = (x² + 4x - 5)(x+3)

Further expanding, we get:

(x² + 4x - 5)(x+3) = x³ + 7x² + 7x - 15

This expanded form helps us identify the coefficients of the polynomial and its degree (which is 3, indicating a cubic function).

Graphing the Function

The graph of the cubic function y = x³ + 7x² + 7x - 15 exhibits several key characteristics:

Intercepts: The function intersects the x-axis at the points (1, 0), (-5, 0), and (-3, 0), as predicted by the factored form. It also intersects the y-axis at (0, -15).
Turning Points: A cubic function typically has two turning points. In this case, we can observe a local maximum and a local minimum.
End Behavior: As x approaches positive infinity, y also approaches positive infinity. Conversely, as x approaches negative infinity, y approaches negative infinity.

Real-World Applications

Cubic polynomials find applications in various fields:

Physics: They can model projectile motion and the behavior of waves.
Engineering: Cubic functions are used in designing structures, analyzing fluid flow, and optimizing processes.
Economics: They can model the growth of economies and the demand for certain goods.

Conclusion

The cubic polynomial (x-1)(x+5)(x+3) offers a fascinating insight into the world of polynomial functions. Its factored form reveals its roots, while its expanded form provides a deeper understanding of its coefficients and degree. Its graph exhibits characteristic features, and its applications are diverse, making it a significant tool in various disciplines.