Exploring the Polynomial (x+1)(x+3)(x+5)^2
This article delves into the fascinating properties of the polynomial (x+1)(x+3)(x+5)^2. We will analyze its structure, explore its roots, and discuss how to find its expanded form.
Understanding the Structure
The polynomial is presented in a factored form, which reveals key information:
 Roots: The factored form immediately tells us the polynomial's roots: 1, 3, and 5 (with 5 having a multiplicity of 2). This means the polynomial will equal zero at these values of x.
 Degree: The polynomial is of degree 4, which we determine by adding the exponents of each factor: 1 + 1 + 2 = 4. This indicates the polynomial will have a maximum of 4 turning points.
Finding the Expanded Form
To understand the polynomial's behavior more clearly, we can expand it:

Expand the square: (x+5)^2 = x^2 + 10x + 25

Multiply the remaining factors: (x+1)(x+3)(x^2 + 10x + 25)

Distribute and simplify: This process involves carefully multiplying each term in one factor by every term in the other factors. This results in a polynomial of the form:
x^4 + 18x^3 + 97x^2 + 175x + 75
Analyzing the Polynomial's Behavior
 Roots: As previously mentioned, the polynomial has roots at x = 1, x = 3, and x = 5 (with multiplicity 2).
 Turning Points: The polynomial is of degree 4, suggesting a maximum of 4 turning points. However, due to the repeated root at 5, it's likely the polynomial will have fewer turning points.
 End Behavior: Since the leading term is x^4, which has an even degree and a positive coefficient, the polynomial will rise on both ends.
Applications
Understanding the properties of polynomials like this is crucial in various fields:
 Calculus: We can use derivatives to analyze the polynomial's critical points, including its maximum and minimum values.
 Algebra: Factoring and expanding polynomials are essential skills in solving equations and inequalities.
 Physics and Engineering: Polynomials often model realworld phenomena, such as projectile motion or the behavior of electrical circuits.
Summary
(x+1)(x+3)(x+5)^2 represents a polynomial with unique characteristics. By analyzing its factored form, we can determine its roots, degree, and potential turning points. Expanding the polynomial provides us with its explicit equation, enabling further exploration of its behavior and potential applications.