Problem 14 - Introductory Algebra for College Students

Question

Factor each difference of two squares. $$x^{4}-25$$

Step-by-Step Solution

Answer

The factored form of \(x^{4}-25\) is \((x^2 + 5)(x+\sqrt{5})(x-\sqrt{5})\)

Step 1: Identify

Identify 'a' and 'b' such that the given expression can be written in the form of \(a^2 - b^2\). Here \(a = x^2\) and \(b = 5\), as \(x^{4} = (x^{2})^{2}\) and \(25 =5^{2}\). So our expression can be written as \((x^{2})^{2} - 5^{2}\).

Step 2: Apply the Difference of Squares formula

Apply the difference of squares formula, \((a^2 - b^2) = (a + b)(a - b)\), to the expression. By substituting \(a = x^{2}\) and \(b = 5\), the equation becomes \((x^2 + 5)(x^2 - 5)\).

Step 3: Factor further if possible

Looking at the factors, \((x^2 + 5)\) cannot be factored further because it is a sum of squares. However, \(x^2 - 5\) is again a difference of squares and should be factored further. Factoring \(x^2 - 5\) yields \((x+\sqrt{5})(x-\sqrt{5})\).

Key Concepts

Factoring

Factoring is a fundamental concept in algebra where we break down an expression into a product of simpler expressions or factors. In the given exercise, you encounter the **difference of squares**, a special case that makes factoring straightforward. A difference of squares looks like this: \[ a^2 - b^2 = (a + b)(a - b) \]The formula expresses the idea that when you subtract one square number from another, you can factor it into two binomials. This is exactly what was done in the exercise:

• The expression \(x^4 - 25\) rewritten as \((x^2)^2 - 5^2\), following the format \(a^2 - b^2\).
• The difference of squares formula was then applied to yield \((x^2 + 5)(x^2 - 5)\).
• Further factoring of \(x^2 - 5\) provided more basic factors \((x + \sqrt{5})(x - \sqrt{5})\).

Factoring expressions helps simplify calculations and solve equations more easily by revealing the underlying structure of polynomials.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operations (such as addition and multiplication). **Understanding** how to manipulate these expressions is key in algebra. In our original exercise, the expression \(x^4 - 25\) is an example of a more complex algebraic expression. It includes:

• The variable \(x\), which can represent any number.
• The exponents, with \(x^4\) which indicates \(x\) multiplied by itself four times.
• A constant, the number 25 in this case.

These elements combine to form the expression, and using operations like subtraction and factoring, we manipulate it to discover more about its structure and solutions. By rewriting and factoring, complex expressions can become manageable, and we can solve or simplify them for various mathematical applications.

Polynomials

Polynomials are algebraic expressions that consist of one or more terms. Each term includes a constant coefficient, variables raised to whole-number exponents. The given expression, \(x^4 - 25\), is a polynomial with two terms (also known as a binomial), specifically crafted into a **difference of squares**. Polynomials can take various forms:

• **Monomials** - a single term.
• **Binomials** - two terms like our example \(x^4 - 25\).
• **Trinomials** - three terms, and so on.

In solving the exercise, the polynomial \(x^4 - 25\) was first recognized as a difference of squares, then decomposed through factoring based on its structure. Understanding the nature of polynomials helps to identify when special factoring techniques, like the difference of squares, can be applied. This is invaluable for simplifying expressions and solving polynomial equations.