Expanding 2 - x^2 in Ascending Powers of x: A Comprehensive Guide

In this article, we will explore the process of expanding the expression 2 - x^2 in ascending powers of x up to and including the term in x^2. This topic is relevant to numerous mathematical applications, such as series expansions, polynomial functions, and optimization problems. By the end of this guide, you will understand the methods and techniques used to perform this expansion and how to apply them in various contexts.

Direct Expansion Method

The simplest way to expand the expression 2 - x^2 is to rewrite it directly and then organize the terms in ascending powers of x. Let's go through the steps:

Write the expression: 2 - x^2 Recognize that x^2 is already in the required form and can be brought to the front. Group the constant term (2) and the x^2 term together: 2 - x^2 4 - 4x x^2

The expression 2 - x^2 can be expanded to 4 - 4x x^2 by considering the constant term and the binomial expansion.

Binomial Expansion Formula

The binomial expansion is a powerful technique for expanding expressions of the form (a b)^n. In this case, we can use the formula for (a - b) where a 2 and b x:

The formula for (a - b)^2 is:

(a - b)^2 a^2 - 2ab b^2

Substituting a 2 and b x:

(2 - x)^2 2^2 - 2(2)(x) x^2

Simplifying the expression:

(2 - x)^2 4 - 4x x^2

Therefore, the expansion of 2 - x^2 in ascending powers of x up to the term in x^2 is:

2 - x^2 4 - 4x x^2

FOIL Method for Multiplication

The FOIL method (First, Outer, Inner, Last) is a technique used to multiply two binomials. Let's apply it to expand 2 - x^2:

First terms: 2 * 2 4 Outer terms: 2 * (-x) -2x Inner terms: -x * 2 -2x Last terms: -x * (-x) x^2

Summing all the terms:

4 - 2x - 2x x^2

Combining like terms:

4 - 4x x^2

Alternative Methods and Observations

In the provided examples, there are some observations and alternative methods that can be discussed:

A - B^2

The expression 2 - x^2 can also be viewed as (2)^2 - (x)^2. Using the difference of squares formula, we have:

(a - b)^2 a^2 - 2ab b^2

Substituting a 2 and b x:

(2 - x)^2 (2^2 - (x)^2) 4 - x^2

However, to include the linear term, we need to fully expand it:

(2 - x)^2 4 - 4x x^2

Ab^2 A^2 - 2ab B^2

Another way to approach this is to use the identity a^2 - 2ab b^2 (a - b)^2. Substituting a 2 and b x:

(2 - x)^2 (2 - x)^2 4 - 4x x^2

Thus, the expansion using (a - b)^2 is the same as using the direct expansion method.

Conclusion

In summary, there are several methods to expand 2 - x^2 in ascending powers of x up to the term in x^2. The direct expansion method, binomial expansion formula, and the FOIL method provide straightforward ways to achieve this. Understanding these methods can help in solving more complex polynomial expressions and in performing mathematical analyses effectively.