Understanding and Simplifying Polynomial Expansions: The Example of (1 - x x^2)^4

In mathematics, polynomial expansions are a fundamental concept in algebra. This article will guide you through the process of expanding the polynomial (1 - x x^2)^4. We will break down each step and explain the methodology used in each stage of the computation.

Introduction to Polynomial Expansion

A polynomial expansion is the process of expressing a given polynomial as a sum of terms. Each term in the expanded polynomial represents the contributions of the product of the original polynomial factors. This process can be quite complex, especially when dealing with higher powers and multiple factors.

Breaking Down the Problem

We start with the polynomial expression (1 - x x^2)^4. This can be expanded by considering it as a product of four identical factors, each being (1 - x x^2). We will tackle the expansion step by step using the distributive property, common in algebra.

Step-by-Step Expansion

Step 1: Write the Power as a Product

The first step is to express (1 - x x^2)^4 as a product of four factors, each being (1 - x x^2): (1 - x x^2) (1 - x x^2) (1 - x x^2) (1 - x x^2)

Step 2: Focus on the Product of the First Two Factors

First, we multiply the first two factors together: (1 - x x^2) (1 - x x^2) 1 - 2x x^2 x^2 x^4 x^4 - 2x^3 3x^2 - 2x 1

List the terms as follows: x^4 - 2x^3 3x^2 - 2x 1.

Step 3: Multiply the Result by the Next Factor

Now, we multiply the result from Step 2 by the next factor, (1 - x x^2): (x^4 - 2x^3 3x^2 - 2x 1) (1 - x x^2) x^6 - 3x^5 6x^4 - 7x^3 6x^2 - 3x 1

Step 4: Distributive Property Application

Next, we apply the distributive property to the product of the first three factors: (x^6 - 3x^5 6x^4 - 7x^3 6x^2 - 3x 1) (1 - x x^2) x^8 - 4x^7 1^6 - 16x^5 19x^4 - 16x^3 1^2 - 4x 1

Step 5: Combine Like Terms

Finally, we combine all like terms in the expanded expression to get the final result: x^8 - 4x^7 1^6 - 16x^5 19x^4 - 16x^3 1^2 - 4x 1

The final expression contains 17 terms.

Conclusion

The process of expanding the polynomial (1 - x x^2)^4 involves breaking down the problem into smaller parts, applying the distributive property, and combining like terms. This method can be extended to more complex polynomial expansions, making it a valuable tool for students and mathematicians alike.

Understanding and simplifying polynomial expansions is crucial for many areas of mathematics, including algebra, calculus, and theoretical physics. By mastering this technique, you can solve a wide range of mathematical problems more efficiently.