Problem 13 - Introductory Algebra for College Students

Question

Factor each difference of two squares. $$x^{4}-9$$

Step-by-Step Solution

Answer

The factored form of \(x^{4} - 9\) is \( (x^{2} - 3)(x^{2} + 3) \).

Step 1: Understand the formula for the Difference of Squares

The formula for the difference of squares is \(a^2 - b^2 = (a - b)(a + b)\), where 'a' and 'b' are the two terms. This formula can be applied to the problem \(x^{4}-9\).

Step 2: Define 'a' and 'b'

In this instance 'a' can be defined as \(x^{2}\) and 'b' as 3, because \((x^{2})^{2}\) is \(x^{4}\), which is a square, and \(3^{2}\) is 9, which is a square as well. The equation now is \(a^{2} - b^{2}\), and can be filled with \(a = x^{2}\) and \(b = 3\).

Step 3: Apply the Difference of Squares formula

Substitute \(a\) and \(b\) into the formula \(a^{2} - b^{2} = (a - b)(a + b)\). This would become \((x^{2} - 3)\) \((x^{2} + 3)\), which is the factorization of the original problem \(x^{4}-9\).

Key Concepts

Factoring Polynomials

Factoring polynomials means breaking down a complex polynomial into simpler components called factors. This process is a fundamental tool in algebra often used to simplify expressions or solve equations. When dealing with certain types of polynomials, like the difference of squares, specific algebraic techniques can be applied effectively.

For example, consider the polynomial given in the exercise:

• It is a powerful tool because it allows you to express an expression in terms of multiplication, which is easier to manage than addition or subtraction.
• Furthermore, factored form gives insight into the roots or solutions of polynomial equations.
• Polynomials can often be factored using patterns like the difference of squares, simplifying problems significantly.

Recognizing and applying these patterns, such as in the equation \( x^4 - 9 \), transforms and simplifies it into controllable units, making the math a whole lot easier.

Algebraic Expressions

Algebraic expressions are like the language of algebra, consisting of numbers, variables, and operations. Understanding how to manipulates these elements is crucial to solving problems efficiently.

In the exercise at hand, we deal with an algebraic expression that contains both a square term and variables, i.e., \( x^4 - 9 \). This is called a polynomial, and it needs to be simplified using special techniques:

• Identify terms: The terms in an expression are defined by operations like addition and subtraction. In \( x^4 - 9 \), there are two terms, \( x^4 \) and \(-9\).
• Apply appropriate algebraic techniques: For certain patterns, like the difference of squares, special formulas directly help in simplifying the expressions.

Understanding expressions this way makes it easier to manipulate and simplify them into familiar forms. This foundational knowledge is vital for anyone tackling more advanced algebraic challenges.

Algebraic Formulas

Algebraic formulas provide shortcuts for factoring and solving intricate problems. They are derived mathematical expressions that help in applying constant operations to varying numbers or variables.

In this exercise, the difference of squares formula is essential. The formula \( a^2 - b^2 = (a - b)(a + b) \) shows that a polynomial of this form can be broken down into a simple product of two binomials:

• The terms \(a\) and \(b\) in this context are \(x^2\) and \(3\) respectively as they align with the squares \(x^4\) and \(9\).
• Applying the formula simplifies the polynomial \( x^4 - 9 \) into \((x^2 - 3)(x^2 + 3)\).

This application not only streamlines the calculation but is a prime example of harnessing algebraic power to break down polynomials. Mastery of such formulas is a significant step in advancing algebra skills.