Problem 65 - College Algebra and Calculus: An Applied Approach

Question

Completely factor the expression.\(\left(x^{2}+1\right)^{2}-4 x^{2}\)

Step-by-Step Solution

Answer

The expression \(\left(x^{2}+1\right)^{2}-4 x^{2}\) completely factors to \((x^{2}+2x+1)(x^{2}-2x+1)\)

Step 1: Identify the perfect squares

In the expression \(\left(x^{2}+1\right)^{2}-4 x^{2}\), the perfect squares are \(\left(x^{2}+1\right)^{2}\) and \(4x^{2}\), where \(4x^{2}\) can be written as \((2x)^{2}\). Thus, the expression can be seen as a difference of squares.

Step 2: Apply the difference of squares formula

We apply the difference of squares formula \(a^{2}-b^{2}=(a+b)(a-b)\) to the expression. So, \((x^{2}+1+2x)(x^{2}+1-2x)\).

Step 3: Simplify each factor

The resulting factors from the difference of squares can be rearranged and simplified to achieve a more standard look. These factors simplify to \((x^{2}+2x+1)(x^{2}-2x+1)\).

Key Concepts

Perfect Squares

Understanding perfect squares is crucial in many algebraic processes, including factoring. A perfect square is a number or algebraic expression that can be written as the square of another number or expression. In other words, it's the product of a number multiplied by itself.

For example, 9 is a perfect square because it can be written as \(3^2\). In algebra, \(x^2\) is also a perfect square because it represents \(x\) multiplied by \(x\). Recognizing perfect squares enables you to identify potential candidates for the difference of squares technique in polynomial factoring. Mastering perfect squares is not just about memorizing a list; it's about understanding the concept of squaring and applying it to both numerical and algebraic expressions.

Difference of Squares Formula

The difference of squares formula is a tool that simplifies the process of factoring certain types of algebraic expressions. This formula, \(a^2 - b^2 = (a + b)(a - b)\), is derived from the pattern observed when you expand the product of two binomials that are conjugates of each other — that is, they have the same terms but opposite signs.

The use of this formula is pivotal in breaking down more complex expressions into simpler, more manageable factors. To apply it, one must identify terms in the expression that can be represented as squares and that are subtracted from one another. Using this formula not only simplifies expressions but also lays the foundation for solving various algebraic equations.

Polynomial Factoring

When it comes to polynomial factoring, it involves breaking down a polynomial expression into a product of simpler polynomials. The result is similar to factoring numbers, where the goal is to express the number as a product of its prime factors. Factoring polynomials is an essential skill in algebra because it is used in simplifying expressions, dividing polynomials, and solving polynomial equations.

There are several factoring techniques, and the difference of squares is one of the more straightforward methods. Polynomial factoring often requires recognizing patterns, and each factoring method responds to a particular pattern. It may also incorporate the use of other methods like the greatest common factor (GCF) or trinomial factoring when dealing with more complex polynomials.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain numbers, variables (like \(x\) or \(y\)), and operation symbols (such as \(+\), \(–\), \(×\), and \(÷\)). These expressions are the building blocks of algebra and form the basis of equations and functions. Unlike equations, algebraic expressions do not have an equals sign.

Learning how to manipulate these expressions through operations such as simplification, expansion, and factoring is a key aspect of algebra. The ability to transform and rewrite algebraic expressions in various forms allows for a deeper understanding of mathematical relationships and is critical for solving complex problems.