Problem 24 - Introductory Algebra for College Students

Question

Factor each difference of two squares. $$x^{4}-1$$

Step-by-Step Solution

Answer

The factored form of \(x^{4}-1\) is \((x^{2}+1)(x+1)(x-1)\).

Step 1: Identify the difference of squares

The given expression, \(x^{4}-1\), can be rewritten as \((x^{2})^2 - 1^{2}\) in order to identify it as a difference of squares where \(a = x^2\) and \(b = 1\).

Step 2: Apply the difference of squares formula

Applying the formula, \((x^{2} +1)(x^{2}-1)\) is the result of factorization. It is important to note, \(x^2 - 1\) is another difference of squares (\(x^2 - 1^2\)) and it can be factored again using the same formula.

Step 3: Apply the difference of squares formula to x^2 - 1

Applying the difference of squares formula to \(x^2 - 1\), we get \((x+1)(x-1)\), thus the whole factorization of the original expression \(x^{4}-1\) becomes \((x^{2}+1)(x+1)(x-1)\)

Key Concepts

Algebraic Expressions

In algebra, an expression is a combination of symbols that is well-formed according to the rules of the discipline. An algebraic expression can consist of constants, variables, and operations such as addition, subtraction, multiplication, division, exponentiation, and root extraction. For instance, the algebraic expression (x^{4}-1) from our exercise involves a variable x raised to the fourth power, which demonstrates its polynomial nature, and a subtraction of 1, a constant. This form of expression is the starting point for many algebraic procedures, including factoring.

Polynomials are a special type of algebraic expression that exclusively uses operations of addition, subtraction, and non-negative integer exponents of variables. Factoring polynomials is a key skill in algebra, which allows us to simplify expressions, solve equations, and understand the properties of functions. In the case of (x^{4}-1), we seek to express it as a product of factors, where each factor is also a polynomial. The factorization process lends itself to identifying patterns, such as the 'difference of squares', which is apparent in our given expression.

Polynomial Factorization

Polynomial factorization refers to expressing a polynomial as a product of its factors. Factors are polynomials themselves that, when multiplied together, give back the original polynomial. The exercise provided (x^{4}-1) is an example where polynomial factorization is applied.

• Difference of Squares: A common pattern in polynomial factorization is the difference of squares. This occurs when a polynomial is in the form of (a^2 - b^2), which can be factored into (a + b)(a - b).
• Iterative Factorization: Sometimes, after applying a factorization pattern, the resulting polynomial can be factored further. This is evident in our problem where (x^2 - 1), a factor of our original polynomial, is also a difference of squares and can be further factored into (x + 1)(x - 1).

By recognizing these patterns, like the difference of squares, we can simplify complex polynomials into products of simpler binomials or other polynomials. This simplification is especially useful in solving polynomial equations and understanding the functions they define.

Basic Algebra Concepts

Basic algebra concepts are the fundamental building blocks for studying and understanding more complex mathematical ideas. Among these are:

• Variables: Symbols like x that represent numbers. They allow algebraic expressions to generalize math truths.
• Exponents: Shows how many times a number or variable is multiplied by itself, for example x^4 indicates x multiplied by itself four times.
• Polynomials: Algebraic expressions that include variables raised to whole number exponents, like the given exercise (x^4 - 1), which is a polynomial of degree four.
• Factorization: The process of breaking down a complex expression into simpler parts (factors) that, when multiplied together, give the original expression. This is exemplified by the step-by-step solution of our difference of squares exercise.

Understanding these concepts is critically important because they form the foundation of algebraic operations and problem-solving. Recognizing differences of squares and other patterns allows one to solve equations more efficiently and aids in the understanding of graphical representations of functions.